partitioned hp-d approach for multiscale transient heat problems - … · 2017-09-19 · element...

Faculty of Civil Engineering and Geodesy

Chair for Computation in Engineering

Prof. Dr. rer. nat. Ernst Rank

Partitioned hp-d approach

for multiscale transient heat problems

Nina Korshunova

Master’s thesis

for the Master of Science program Computational Mechanics

Author: Nina Korshunova

Matriculation number: 03658242

Supervisor: Prof. Dr.rer.nat. Ernst Rank

Dr.-Ing. Stefan Kollmannsberger

Nils Zander, M.Sc.

Date of issue: 26. March 2016

Date of submission: 26. September 2016

Involved Organisations

Chair for Computation in EngineeringFaculty of Civil Engineering and GeodesyTechnische Universitat MunchenArcisstraße 21D-80333 Munchen

Declaration

With this statement I declare, that I have independently completed this Master’s thesis. Thethoughts taken directly or indirectly from external sources are properly marked as such. Thisthesis was not previously submitted to another academic institution and has also not yetbeen published.

Munchen, September 25, 2016

Nina Korshunova

Nina Korshunovae-Mail:[email protected]

Acknowledgments

I would like to express my gratitude to many people, who made the realization of this thesispossible.

First of all, I would like to thank Nils Zander for the amount of time and energy, he investedin this thesis. I am really grateful, that I got a chance to learn from such an inspiring person.With his invaluable advices and ideas he made a great contribution to this work. His pos-itive attitude and very structured approach helped me to organize this thesis and complete it.

I would like to thank Dr.-Ing. Stefan Kollmannsberger for the accurate and successful leadingof this work. I am grateful for all the discussions we had, which led to new ideas and newchallenges. Moreover, I want to thank him for the amazing atmosphere he created for meand for the attitude, which made my work easy, enjoyable and possible.

I am thankful to Prof. Ernst Rank, who gave me the inspiration for this topic, who sup-ported me through all my thesis. I want to thank him for supervising and examining my work.

I would also like to take this opportunity to thank Philipp Kopp and the entire research groupof the Chair for Computation in Engineering. I am grateful for the endless debugging ses-sions, for the help in the derivations for this work and for the constructive review of my thesis.

Special thanks are due to Inigo Lopez. His invaluable everyday support, care and under-standing helped me all along the way. I would like to thank him for his constant reviewingand contribution to this thesis. His ideas and suggestions had a big influence throughout allmy work.

Thanks to my fantastic friends for being there for me in each stage of my life. I know thatreading this thesis for them was really a lot of fun. I am grateful for their patience and alltheir creative ideas,which motivated me while writing this thesis.

Finally, I want to thank my family. I know that we see each other only once or twice peryear, but they were there for me through all my life. They should know that without themnone of these would be possible.

VII

Contents

1 Introduction 1

1.1 Motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Finite Element Method in heat transfer analysis 5

2.1 Strong form of the nonlinear transient heat problem . . . . . . . . . . . . . . 5

2.2 Weak form of the heat problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Galerkin approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 Hierarchical basis functions . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.3 L2 projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.4 Gaussian quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 One-step algorithms for semi discrete heat problems . . . . . . . . . . . . . . 12

2.5 Iterative algorithms for nonlinear heat problems . . . . . . . . . . . . . . . . . 13

3 Refinement schemes 17

3.1 Replacement refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Superposition refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 The multiscale hp-d method for heat transfer problems 23

4.1 Linear stationary heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Overlay approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.2 Implementational aspects . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.4 Significance of the coupling terms . . . . . . . . . . . . . . . . . . . . . 34

4.2 Linear transient heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


4.2.2 Time discretization of the overlay approximation . . . . . . . . . . . . 37

4.2.3 Implementation of hp− d method . . . . . . . . . . . . . . . . . . . . 41

4.2.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.5 Significance of the coupling terms . . . . . . . . . . . . . . . . . . . . . 49

4.3 Nonlinear stationary heat equation . . . . . . . . . . . . . . . . . . . . . . . . 50


4.3.2 Implementation of the hp− d method . . . . . . . . . . . . . . . . . . 63

4.3.3 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.4 Significance of the precision of the Jacobian computation and the cou-pling terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Nonlinear transient heat equation . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.1 Overlay approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.2 Time discretization of the overlay approximation . . . . . . . . . . . . 954.4.3 Implementation of hp− d method . . . . . . . . . . . . . . . . . . . . 954.4.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.5 Significance of the precision of the Jacobian computation and the cou-

pling terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Summary and Conclusions 107

A Linear stationary heat equation 111

B Linear transient heat equation 115

C Nonlinear stationary heat equation 119

D Nonlinear transient heat equation 129

E Compact disc 137

1

Chapter 1

Introduction

1.1 Motivation and goals

Additive manufacturing (AM) is a novel and fast-developing technology. A free-shape three-dimensional object can be created by adding layers through non-melting or melting technol-ogy. The main advantage of this method is the production of complex geometries with highaccuracy and minimum material waste. Moreover, the small parts, which by conventionalmanufacturing cannot be monolithically created, nowadays can be consolidated into a singleobject, improving its functionality and mechanical properties [Huang et al., 2013]. Additivemanufacturing becomes attractive for industries such as aerospace, automotive, electronicsand biomedicine. The freedom and complexity of possible designs allow fast prototypingwithout a specific tool, while the fabrication costs are reduced.

Two of the most popular methods in additive manufacturing are the selective laser sintering(SLS) and selective laser melting (SLM). In the SLS process the powder is added layer-by-layerafter sintering the previous one [Lu et al., 2001]. The information about the two-dimensionalprofile is provided by the CAD data. The SLS process provides full control over the porosityof the final product. The SLM technology is similar to the one described before, differingonly by the stage of the heat used for the material powder. The SLS process does not fullymelt the powder but heats it to the point of fusion at the molecular level. In contrast, theSLM process melts the powder by using higher laser energy density and therefore allows thecreation of the fully dense parts with the material properties close to the bulk material [Levyet al., 2003].

As higher energy densities are involved in the SLM process, the main defects of the finalproduct are porosity, residual powder and non-connected layers [Attar, 2011]. The qualityof the outcome is dependent on the used material, size and geometry of the produced spec-imen. Thus, the estimation of the process variables prior to the production stage is needed.The modern optimization is performed through an expensive trial and error process. There-fore, the numerical simulation of the additive manufacturing process can better quantify theinvolved physics and improve the control over the production machines.

The simulations of the SLM processes via the conventional Finite Element Method are veryexpensive and complex, which can be attributed to the physical phenomena of the additivemanufacturing. First, the single powder particles in the zone of the laser effect are in the

2 1. Introduction

melted phase, while the last applied layer is already cooling down. Fast cooling and heat-ing rates lead to the redistribution of the stresses, which causes the development of residualstresses and deformations through the whole specimen affecting geometrical accuracy [Neuge-bauer et al., 2014]. In order to predict this physical behavior through numerical simulationthe multi-phase models should be applied on the different scales. As the phases are highlydependent on the current temperature level, the nonlinearity of the material is governingthe additive manufacturing process. Moreover, the high-speed traveling of the laser beamthrough the domain increases the complexity of the numerical computation, as the wholedomain should be resolved at each time step up to the smallest scales. Furthermore, thesize of the laser is orders of magnitudes smaller than the size of the working specimen. Tocapture the high gradient temperatures around the laser the multi-scale models are thereforenecessary.

In order to reduce the computational effort, two main research areas exist: the developmentof new physical models describing the physical behavior and the improvement of the existingnumerical approaches by for example introducing adaptive (re-)meshing algorithms. Themost popular approach within the first area is the use of two-dimensional models, which lackaccuracy when applied to complex geometries and do not allow to account for non-connectedlayers [Daneshgar et al., 2016]. An Eulerian approach developed by Zhang and Michleris is toreduce the computational effort, yet leading to a precision loss compared to the Lagrangianformulation [Zhang and Michaleris, 2004]. The second area of research is in the investigationof the computational costs using adaptive meshing algorithms. This approach uses an errorestimator in order to evaluate the region of interest and refine it, while coarsening the rest ofthe domain. First, the studies on this have been performed in the context of the interactionof a shock with a dense cloud for finite differences, as in [Bell et al., 1994]. More research wascarried out in terms of multi-level hp−adaptivity, introduced in [Zander et al., 2015]. Thisapproach will be used as a reference in the current work.

The main focus of this thesis is the development of a special mesh refinement technique,allowing the reduction of the numerical effort when applied to the simulations of additivemanufacturing process.

Different refinement techniques are investigated to numerically simulate multiscale problems.The s − method as introduced by J. Fish [Fish et al., 1994] is using the superposition ofindependent finite element meshes. The domain decomposition methods were also analyzedby Bramble [Bramble et al., 1990]. In the present thesis the multiscale hp − d domaindecomposition is used. At first, the concept of hp − d refinement was introduced by Prof.Rank [Rank, 1992b]. The concept of this method underlays in the domain decomposition toglobal and local scales, which can be solved separately, accounting for the coupling betweentwo domains. The hp − d method was studied in the application to linear problems [Rank,1993], [Rank and Krause, 1997], [Duster et al., 2007]. The concept was also applied tononlinear models using the Finite Cell Method [Schillinger et al., 2010].

As the hp − d method was not applied to nonlinear transient problems within the FiniteElement concept, The aim of this work is to extend the hp − d concept to the regime ofnonlinear transient problems. This approach will be addressed in the present thesis. Thisapplication will allow to decrease the computational costs of the additive manufacturingprocess due to the ability of simulating local phenomena separately from the large scale partof the specimen. Moreover, there is no limitation concerning the introduction of new materialmodels for the small scale zone (for example, to capture the melting phase). Within the frame

1.2. Outline 3

of transient computations the introduced time sub stepping for the small scale phenomenawill allow more precise computation of high gradients, while keeping the numerical effort atthe same level. In the scope of the current work the first step in this direction, focusing onone-dimensional problems, is made.

1.2 Outline

In order to give an outline of this thesis, the content of each chapter will be briefly explainedin this section.

First, the governing equations of the nonlinear heat problem is introduced in Chapter 2.From the strong form of the physical phenomena, the weak form, imposing lower smoothnessrequirements on the solution field, is derived in Section 2.2. After the mathematical modelfor the nonlinear transient heat flow is defined, the concept of the Finite Element Method isintroduced and the numerical model is derived in Section 2.3. For the convenience of furtherexplanations, some details about Finite Element Method are given, such as an introduction tothe hierarchical basis functions (see Subsection 2.3.2), the L2 projection (see Subsection 2.3.3)and the Gaussian quadrature (see Subsection 2.3.4). As transient problems are consideredin the scope of current work, the algorithms for semi-discrete heat problems are introducedin Section 2.4. For nonlinear problems, iterative algorithms are widely used to obtain thesolution of the numerical model. This is introduced in Section 2.5.

This work deals with a special superposition refinement technique, the partitioned multi-scale hp − d method. In order to give an overview of existing methods, replacement andsuperposition refinement techniques are described in Chapter 3. The hp− d approach, whichwill be developed in the partitioned manner in the present thesis, is introduced in detail inSection 3.2. All the results are compared to the multilevel hp−approach, implemented inthe in-house high-order FEM code AdhoC++. Therefore, an introduction to the referencemethod is provided in Section 3.2.

Chapter 4 shows applications of the partitioned multiscale hp − d approach to the linearstationary (Section 4.1), linear transient (Section 4.2), nonlinear stationary (Section 4.3) andnonlinear transient (Section 4.4) one-dimensional heat equations. Each section is structured inthe same manner. First, the partitioned overlay approximation is applied to the mathematicalmodel and the numerical model is developed. Then, implementational aspects are described.Lastly, a verification of the developed code with AdhoC++ is performed. Furthermore, as thefurther implementation costs of the newly developed hp − d approach should be estimated,the significance of the coupling terms is evaluated.

The work concludes with the summary and outlook in Chapter 5.

4 1. Introduction

5

Chapter 2

The Finite Element Method in heattransfer analysis

In additive manufacturing the governing equations for the physical model of the process areheat equations. Therefore, in this chapter the finite element method is applied to heat transferproblems. At first, strong and weak forms for the general non-linear transient heat problemare stated, followed by the introduction of the Finite Element approximation. In this section,the aspects of the Finite Element formulation, which are going to be used in later sections,are described further.

2.1 Strong form of the nonlinear transient heat problem

The main elements for modeling the physical phenomena are:

• The partial differential equation (PDE), which governs the physical quantities throughthe domain

• Boundary conditions, which characterize the interaction between the physical domainand the external environment

• Initial conditions, which describe the physical quantity at a fixed point in time.

Together, these three components form a well posed initial boundary value problem whenboundary and initial conditions are chosen correctly. In reality, most physical properties of theheat transfer problem are temperature-dependent. Therefore, the nonlinear physical modelis necessary. Moreover, as the processes are usually transient, the model should account forthose effects. The general differential equation for a nonlinear transient heat problem is:

ρ(T )c(T )∂T

∂t−∇ · (k(T )∇T ) = f in Ω, (2.1)

where Ω is the considered domain, T is the scalar temperature distribution, ρ represents thedensity of the used material, c is the heat capacity, t defines the time, ∇ is the gradient

6 2. The Finite Element Method in heat transfer analysis

operator[∂∂x

∂∂y

∂∂z

], k stands for the thermal conductivity and f is the heat supply.

For the complete formulation of the problem the boundary conditions need to be appliedon the whole boundary of the domain Γ = ∂Ω, which comprises the Dirichlet ΓD and theNeumann part ΓN with ΓN ∩ΓD = 0. They can be of two types: essential, which are definedon the Dirichlet boundary ΓD, and natural boundary conditions defined on the Neumannboundary ΓN (see Figure 2.1).

natural BC: ΓN

essential BC: ΓD

x

y

z

Figure 2.1: Type of the boundary conditions (BCs)

The temperature condition, or essential boundary condition, is prescribed at specific bound-ary ΓD:

T = T0 at ΓD. (2.2)

The heat flow condition, or natural boundary condition, is specified as follows:

−kn∂T

∂n

∣∣∣∣ΓN

= q0 at ΓN, (2.3)

where kn denotes thermal conductivity in the direction of the normal to the surface of theNeumann boundary and q0 is the prescribed heat flux.

Moreover, the convection and radiation boundary conditions can be applied to the heatproblem [Bathe, 2007]. Both of them are considered in Equation 2.3. For the convectionboundary condition, one can write the following formulation of the prescribed boundary heatflux:

2.2. Weak form of the heat problem 7

q0 = h(Te − TΓN ), (2.4)

where h is the convection coefficient, Te is the environmental temperature and TΓN is thetemperature at the boundary ΓN .

The radiation boundary condition is defined as follows:

q0 = κ(Tr − TΓN ), (2.5)

where κ is the radiation coefficient and Tr is the temperature of the external radiative source.

When the transient problem is considered, the initial conditions need to be specified:

T |t=to = Tinit at t = t0. (2.6)

2.2 Weak form of the heat problem

The next step towards the finite element formulation is to obtain the weak form of theconsidered problem. It can be achieved by the variational calculus:

Find T ∈ X such that 〈ρ(T )c(T )T , v〉 −A (T, k(T ), v) = F (v) ∀v ∈ V, (2.7)

where the nonlinear form A (T, k(T ), v) is dependent on the temperature field T , k(T ), thetemperature dependent thermal conductivity and v, the virtual temperature function, 〈·, ·〉 isthe inner product and F (v) is a linear functional [Duster, 2008]. The set of admissible trialsolutions is defined in Equation 2.8 and the set of admissible weighting functions is specifiedin Equation 2.9.

The weighting functions are required to be zero at the Dirichlet boundaries of the domain[Hughes, 1987].

X = T |T ∈ H1(Ω), T = T0 ∀T ∈ ΓD, (2.8)

V = v | v ∈ H1(Ω), v = 0 ∀v ∈ ΓD. (2.9)

Applying the variational calculus one can rewrite Equation 2.1 with the corresponding bound-ary conditions as follows:

∫Ω

ρ(x, T )c(x, T )∂T

∂tv dΩ−

∫Ω

∇ · (k (x, T )∇T ) v dΩ =

∫Ω

f(x) v dΩ. (2.10)

Integrating Equation 2.10 by parts, the weak form of the nonlinear transient heat problembecomes:


∫Ω

ρ(x, T )c(x, T )∂T

∂tv dΩ +

∫Ω

∇vk (x, T )∇T dΩ =

∫Ω

f(x) v dΩ +

∫Γ

k (x, T )∇TvdΓ, (2.11)

whereas the boundary term is zero at the Dirichlet boundary ΓD of the domain Ω and thehomogeneous Neumann boundary ΓN .

2.3 Finite Element Formulation

As the weak form of the governing partial differential equations is known, the finite elementmethod can be applied to obtain the numerical solution of the considered heat boundary-valueproblem.

2.3.1 Galerkin approximation

The Bubnov-Galerkin approximation can be used to obtain the approximate solution of thecontinuous boundary-value problem (Equation 2.7). The discrete form of the weak formula-tion can be stated after choosing finite dimensional subspaces Xh ∈ X and V h ∈ V :

Find T h ∈ Xh such that 〈ρ(T h)c(T h)T h, vh〉 −A (T h, k(T h), vh) = F (vh) ∀v ∈ Vh.(2.12)

The solution and weighting functions are approximated according to the Bubnov-Galerkinmethod by the sum of the shape functions along the element subspaces [Zienkiewicz, 2005]:

v ≈ vh =

ndof∑j=1

Nj vj , (2.13a)

T ≈ T h =

ndof∑i=1

NiTi. (2.13b)

The discretization of Equation 2.11 with Equations 2.13a and 2.13b leads to the followingequation:

2.3. Finite Element Formulation 9

ndof∑j=1

vj

ndof∑i=1

˙Ti

∫Ω

Niρ

(x,

ndof∑i=m

NmTm

)c

(x,

ndof∑i=m

NmTm

)Nj dΩ

+

ndof∑j=1

vj

ndof∑i=1

Ti

∫Ω

∇Nik

(x,

ndof∑m=1

NmTm

)∇Nj dΩ =

=

ndof∑j=1

vj

∫Ω

f(x)Nj dΩ +

ndof∑j=1

ndof∑i=1

[Njk

(x,

ndof∑m=1

NmTm

)∇NiTivj

].

(2.14)

As Equation 2.14 holds for any arbitrary vj , the compact form of the discretized heat equationfor each j = 1...ndof can be obtained by:

ndof∑i=1

Mji

(ndof∑m=1

NmTm

)˙Ti +

ndof∑i=1

Kji

(ndof∑m=1

NmTm

)Ti = fj , (2.15)

or in matrix form:

M(T)T + K(T)T = F, (2.16)

with

M(T) =

∫Ω

NTρ(x,NT)c(x,NT)NdΩ, (2.17)

K(T) =

∫Ω

∇NTk(x,NT)∇NdΩ, (2.18)

F =

∫Ω

NT f(x)dΩ. (2.19)

2.3.2 Hierarchical basis functions

The quality of the approximated solution is dependent on the chosen finite element subspacesXh ∈ X and V h ∈ V , which can be represented by different sets of the shape functionsN . As mentioned previously, the Bubnov-Galerkin method implies the choice of the sameshape functions for the approximation of the solution field and the weighting functions.However, in order to choose these shape functions, the following aspects should be taken intoconsideration:

• Efficient computation of the element matrices


• Efficient enforcement of the exact and minimal required continuity inside and betweenthe elements

• Small round-off error accumulation with respect to the increase of the polynomial degree

• High performance of the iterative solution procedures

• High approximation power

These considerations are fulfilled by the construction of, for example, the orthogonal andhierarchic shape functions [Szabo and Babuska, 1991]. As the hierarchic shape functionsbased on integrated Legendre polynomials lead to the better structure and better conditionnumber of the element system matrices, they are chosen to be the basis for the finite elementsubspaces. Moreover, the main advantage of the hierarchic basis functions is, that lower ordermodes are contained in the high order basis, as the condition number of the element stiffnessmatrices is strongly improved.

The hierarchic shape functions based on integrated Legendre polynomials were first intro-duced in [Szabo and Babuska, 1991]:

N1(ξ) =1− ξ

2(2.20)

N2(ξ) =1 + ξ

2(2.21)

Ni(ξ) = φi−1(ξ) i = 3, 4, ..., p+ 1, (2.22)

with

φj(ξ) =

√2j − 1

2

ξ∫−1

Lj−1(x)dx =1√

4j − 2(Lj(ξ)− Lj−2(ξ)) j = 2, 3, ... , (2.23)

where the integrated Legendre polynomials Lj are:

Lj(ξ) =1

2jj!

dj

dxj(ξ2 − 1)j , j = 0, 1, 2, ... . (2.24)

Figure 2.2 shows the one-dimensional hierarchic basis functions. In order to obtain a two-dimensional hierarchic basis, the tensor product of the respective one-dimensional modes isapplied.

2.3. Finite Element Formulation 11

N1 N2

N3

N4

-0.5 0.5

1.0

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1.0

Figure 2.2: Hierarchic basis functions in one dimension

2.3.3 L2 projection

The initial conditions are necessary to complete the Finite Element Formulation of the con-sidered problem. As they are a part of the continuous formulation of the problem, theprojection on the finite element space is needed. In the scope of this thesis the orthogonal-,or L2-projection is used. In contrast to the interpolation at the nodes, it gives an integral ap-proximation [Larson, 2013]. Mathematically, the L2-projection Phf of a function f is definedas follows:

∫Ω

(f − Phf)vdΩ = 0, ∀v ∈ V h. (2.25)

Now, if the same discretization technique as above is applied, the L2-projection becomes:

Mf = F, (2.26)

where M =∫Ω

NTNdΩ, F =∫Ω

NT fdΩ and f represents continuous function to be projected.

2.3.4 Gaussian quadrature

As the shape functions are chosen, the precise computation of the element matrices is neces-sary. They involve the computation of the integrals of the following type:

I =

∫Ω

F (x, y, z)dΩ. (2.27)


First, the transformation to the standard element space [−1, 1] is determined:

I =

1∫−1

F (Qx(ξ, η, ζ), Qy(ξ, η, ζ), Qz(ξ, η, ζ))‖J‖dξdηdζ, (2.28)

where Qx(·), Qy(·), Qz(·) determine the mapping between Cartesian and natural element co-ordinates, and the Jacobian J represents the derivatives of the Cartesian coordinates withrespect to the natural ones:

J =

∂x∂ξ

∂y∂ξ

∂z∂ξ

∂x∂η

∂y∂η

∂z∂η

∂x∂ζ

∂y∂ζ

∂z∂ζ

. (2.29)

The most common numerical integration used in Finite Element computations is Gaussianquadrature. It is an approximation to the continuous integral defined in the range of [−1, 1]as the weighted sum of the function, evaluated at the specified points, i.e. Gauss points.Therefore, the integral in Equation 2.28 can now be approximated as follows:

I ≈n∑r=1

m∑s=1

k∑l=1

wrwswl (F (Qx(ξ, η, ζ), Qy(ξ, η, ζ), Qz(ξ, η, ζ))‖J‖)ξr,ηs,ζl (2.30)

where n,m, l define the number of the Gauss points in the respective coordinate directions.The factors wr, ws, wl define the Gauss weights.

As the Gauss quadrature rule is only an approximation of the integral, the accuracy ofthe evaluation of the integral highly depends on the number of chosen Gauss points. Thequadrature is exact for polynomials integrals of degree 2n − 1 or less. As not all of thefunctions in the current work will be continuous, the composed integration rule should beoften applied, by dividing the integration domain to the smaller parts, where the function isdefined continuously.

2.4 One-step algorithms for semi discrete heat problems

The semi discrete Equation 2.15 can be discretized in time. The most common family of meth-ods is the generalized trapezoidal family, which consists of the following equations [Hughes,1987]:

Mvn+1 + KTn+1 = Fn+1 (2.31)

Tn+1 = Tn + ∆tvn+α (2.32)

vn+α = (1− α)vn + αvn+1 (2.33)

where Tn+1 and vn+1 are approximations to T(tn) and T(tn) respectively, ∆t is assumed to

2.5. Iterative algorithms for nonlinear heat problems 13

be constant. The coefficient α is dependent on the chosen method (see Table 2.1).

α Method

0 Forward Euler12 Crank-Nicolson1 Backward-Euler

Table 2.1: Methods belonging to the generalized trapezoidal family

As the Backward-Euler scheme is fully implicit, it is used in the scope of the current work.Considering Equations 2.31 - 2.33, Equation 2.15 is solved for ∆T = Tn+1 − Tn:

(M + ∆tK) ∆T = ∆t (Fn+1 −KTn) (2.34)

Tn+1 = Tn + ∆T (2.35)

2.5 Iterative algorithms for nonlinear heat problems

The root finding method Newton-Raphson is the most common approach to solve nonlinearsystems of equations with temperature dependent material parameters [Cottrell et al., 2009].Therefore, Equation 2.15 can be represented in the residual form, which is required to bezero.

R(T) = F−K(T)T−M(T)T!

= 0 (2.36)

First, a linearization of Equation 2.36 is obtained, using Taylor series [Bonet and Wood,2008]:

R(Tk−1 + ∆T) ≈ R(Tk−1) +DR(Tk−1)[∆T] +H.O.T. (2.37)

where DR(Tk−1)[∆T ] denotes the directional derivative of the residual R(Tk−1), evaluatedat the previous iteration k − 1 with respect to the increment of the solution ∆T and k = 1...Nis the Newton-Raphson iteration counter. The terms in Equation 2.37 are now linear, whilethe high order terms are neglected.

As the residual at the next iteration is also required to be zero, Equation 2.37 can be rewrittenas below:

DR(Tk−1)[∆T] = −R(Tk−1). (2.38)

with the update of the temperature field for the next Newton-Raphson iteration step:

Tk = Tk−1 + ∆T. (2.39)


The equation for the residual in index form for each degree of freedom is written as:

Rj =

∫Ω

f(x)Nj dΩ−ndof∑i=1

T k−1i

∫Ω

∇Njk

(x,

ndof∑m=1

NmTk−1m

)∇Ni dΩ

−ndof∑i=1

˙Ti

∫Ω

Njρ

(x,

ndof∑i=m

NmTm

)c

(x,

ndof∑i=m

NmTm

)Ni dΩ.

(2.40)

The directional derivative is a partial derivative with respect to the chosen degree of freedomfor the discretized case. Therefore, the derivative of the respective residual with respect toTi yields:

∂Rj(Ti)

∂Ti=−

∫Ω

∇Njk(x,

ndof∑m=1

NmTk−1m )∇Ni dΩ

−∫Ω

∇Njk′(x,

ndof∑m=1

NmTk−1m )

ndof∑m=1

(∇NmT

k−1m

)Ni dΩ

−∫Ω

Njρ(x,

ndof∑m=1

NmTk−1m )c(x,

ndof∑m=1

NmTk−1m )Ni dΩ

−∫Ω

Njρ′(x,

ndof∑m=1

NmTk−1m )c(x,

ndof∑m=1

NmTk−1m )

ndof∑m=1

(Nm

˙Tk−1

m

)Ni dΩ

−∫Ω

Njρ(x,

ndof∑m=1

NmTk−1m )c′(x,

ndof∑m=1

NmTk−1m )

ndof∑m=1

(Nm

˙Tk−1

m

)Ni dΩ,

(2.41)

or in matrix notation:

DR(Tk−1)[∆T] =(K(Tk−1) + K′(Tk−1)

)∆T+

(M(Tk−1) + M′(Tk−1) + M′(Tk−1)

)∆T,

(2.42)

where

2.5. Iterative algorithms for nonlinear heat problems 15

K(Tk−1) =

∫Ω

∇NTk(x,NTk−1)∇N dΩ =

∫Ω

BTk(x,NTk−1)B dΩ (2.43)

K′(Tk−1) =

∫Ω

∇NTk′(x,NTk−1)(∇NTk−1

)N dΩ =

∫Ω

BTk′(x,NTk−1)(BTk−1

)N dΩ

(2.44)

M(Tk−1) =

∫Ω

NTρ(x,NTk−1)c(x,NTk−1)N dΩ (2.45)

M′(Tk−1) =

∫Ω

NTρ′(x,NTk−1)c(x,NTk−1)(NTk−1

)N dΩ (2.46)

M′(Tk−1) =

∫Ω

NTρ(x,NTk−1)c′(x,NTk−1)(NTk−1

)N dΩ (2.47)

17

Chapter 3

Refinement schemes

For the simulation of the additive manufacturing process, a high resolution of the zone nearthe laser stroke is needed to capture the high gradients of the temperature field. Most of thewell-investigated approaches use classical refinement schemes, such as h− and p− refinement.Though they can represent the solution field well, these methods are computationally expen-sive in the application to multiscale problems. The aim of this thesis is to asses the numericalperformance of the superposition refinement techniques for the additive manufacturing pro-cess simulations. This chapter provides the description of both approaches and introducesthe hp− d refinement technique in detail, which is the focus of this thesis.

3.1 Replacement refinement

Conventional refinement schemes involve the replacement of the region of interest by a subsetof finer elements or high-order elements, which approximate the solution gradients better.Following [Zienkiewicz, 2005], three general groups of the refinement strategies exist:

• h−refinement: The smaller elements are introduced in the region, where better ap-proximation is required, while keeping their order the same in the whole domain. Asin the rest of the domain the elements are coarsened, this approach provides a goodcompromise between the solution quality and the numerical effort.

• p−refinement: The polynomial order of the elements is raised, keeping the same size.This approach leads to more accurate solutions, where the field cannot be representedby the combination of linear polynomials and where the solution is smooth.

• hp−refinement: This approach combines both advantages of the above mentioned meth-ods. Commonly, h−refinement is used next to the discontinuities, while p−refinementin the rest of the domain.

h−refinement

Figure 3.1 shows three different concepts of the most common h−refinement techniques.

18 3. Refinement schemes

a) Original mesh b) Element subdivision

c) Remeshing d) r-refinement

Figure 3.1: Different h−refinement techniques (adopted from [Zienkiewicz, 2005])

The first approach is element subdivision. The existing large elements from the originalmesh, that showed the biggest error, are replaced by the set of the finer elements. As theinitial boundary of those elements is kept the same, the refined elements has the nodesthat do not correspond to any nodes of the non-refined elements. This creates a gap inthe solution through the refinement border. Therefore, these hanging nodes need a specialtreatment as the global convergence without the compatibility of the interconnected elementscannot be guaranteed. The most common approach is to fulfill the compatibility condition byconstraining the hanging nodes in a post-refinement step [Demkowicz, 2006]. Another way ofavoiding hanging nodes could be an introduction of the absent fine-scale modes to the coarseelements [Fries et al., 2011]. The most expensive way of dealing with the hanging nodes isto introduce a smooth transition between the local refinement zone and the coarse elements.It increases the computational costs drastically and removes the local nature of the elementsubdivision [Rivara, 1984].

3.2. Superposition refinement 19

The more computationally costly method of h− refinement is remeshing (see Figure 3.1).When the error of the obtained solution is estimated and the zone of refinement is defined, thenew mesh is created according to the error indicators. Compared to the element subdivision,there are no implementational difficulties with refining and coarsening, as a completely newmesh is created. Nevertheless, the regeneration of the mesh is numerical expensive due tothe transfer of the existing mesh data to the newly created adn difficult to automate.

Another method is r−refinement. The total number of degrees of freedom is kept at thesame level, while moving the nodes in space (see Figure 3.1). However, if the global meshis not accurate enough to capture physical phenomena, this method will not lead to betterapproximation results.

p−refinement

When the global solution is smooth, the p−refinement becomes more efficient. The refinementcan be performed either by raising the polynomial order of the shape functions in the wholedomain, or locally, using hierarchical refinement.

hp−refinement

The combination of the two approaches is also commonly used. This technique becomes veryefficient in the presence of discontinuities (for example, L−shaped domain). In this case, theenrichment with the subset of finer elements is performed close to the discontinuity. In therest of the domain the order of polynomial degree is gradually raised, keeping linear elementsin the region of discontinuity. This approach can give exponential convergence, compared tothe h− and p−refinement separately [Duster, 2008].

3.2 Superposition refinement

The approach, followed in the present thesis, is based on the superposition of a finer overlaymesh and a coarse base mesh. The introduction of a finer resolution allows an improvementof the small-scale solution characteristics.

The research in this area is developing very fast. There are already existing methods, suchas the multigrid method (see e.g. [Trottenberg et al., 2000]), the s−method (see e.g. [Fishand Markolefas, 1993]), the adaptive local overlapping grid method (see e.g. [Moore andFlaherty, 1992]), the Extended or the Generalized Finite element method (see e.g. [Babuskaand Melenk, 1996]) etc. The main concept of all these methods is the decomposition of thefinal solution as the sum of two components: the base and the overlay solution. As thecompatibility of the full solution should be fulfilled, the overlay approximation is normallyconstrained using homogeneous Dirichlet boundary conditions on the transition boundarybetween base and overlay meshes. Therefore, the problem of hanging nodes does not existfor these approaches by construction.

In the current work two groups of methods are described more in detail: the multilevelhp−method and the multiscale hp − d approach. The multiscale approach was studied forthe linear stationary problems (see e.g. [Rank, 1992a], [Rank, 1992b]). Nevertheless, theadditive manufacturing phenomena, introduced before, require nonlinear transient multiscalesimulations with the decrease of the computational costs. Thus, the partitioned hp−d method


becomes extremely attractive. As the multilevel hp−method in a specific case leads to themonolithic version of the hp− d approach, it is explained and used for further verification ofthe developed methodologies.

The multiscale hp− d method

The concept of hp− d approximation was proposed in [Rank, 1992a] and [Rank, 1992b]. Itsidea is visualized in Figure 3.2.

First, the coarse h− or p− base mesh (Ωb) is constructed spanning the whole computationaldomain Ω. The base solution Tb captures the large-scale solution behavior. In order toimprove the base solution Tb locally, an independent fine h−discretization is applied to thedomain of interest (Ωo) (for example, the domain with the highest error in the solutionapproximation). Then the total solution of the system can be determined as a superpositionof the approximation on the base and overlay meshes:

Thp−d = Tb + To (3.1)

basemesh: Tb

overlaymesh: To

hp-d approxi-mation: Thp−d

Figure 3.2: hp− d concept for one-dimensional problems

In contrast to the multilevel hp−approach, multiscale methods allow the solution of two scalesseparately, which leads to the full decoupling of two problems reducing the computational ef-fort. The original idea of the hp−d approach includes only one overlay mesh, restricting it tothe linear modes. Nevertheless, there are no restrictions to the extension of this approach tomore overlay meshes [Schillinger and Rank, 2011] and introduction of the high-order modes.This enables to use different time steps on overlay and base meshes or to use different ma-terial models. In this master thesis only one overlay mesh with linear modes is used, as themethodology has to be developed first regarding the application of fully decoupled nonlinearproblems, which can be later extended to multiple meshes with high-order modes.

Two major aspects have to be considered while constructing overlay and base meshes (seeFigure 3.3) [Krause and Rank, 2003]:

3.2. Superposition refinement 21

• The total solution approximation should be at least C0 − continious between the ele-ments

• No linear dependencies should be presented between the shape functions of the baseand the overlay mesh

Active nodesInactive nodes

Inactive edgesActive edgesActive facesInactive faces

a) 1D case b) 2D case

Figure 3.3: Multiscale hp− d meshes construction (adopted from [Zander et al., 2015])

The first aspect, the compatibility of the hp − d solution approximation, is guaranteed byapplying homogeneous Dirichlet boundary conditions on Γo\ (Γb ∩ Γo), where Γo defines theboundary of the overlay mesh and Γb the boundary of the base mesh.The essential boundarycondition is applied also on the Dirichlet boundary ΓD ⊂ Γb. In case Neumann boundaryconditions are applied to ΓN ⊂ Γb, then this condition also needs to be applied to Γo ∩ Γb[Duster et al., 2007].

The second aspect, the linear independence, can be achieved by the hierarchical nature ofthe overlay elements. The nodal modes of the overlay mesh defined on Ωo, that are directdescendants of the base mesh nodes, are ”deactivated” (see black nodes in Figure 3.3). Thisensures the linear independence between the shape functions of the overlay and the basemeshes [Zander et al., 2015].

The multilevel hp−method

The idea of multilevel hp−method is the combination of high-order hierarchically overlayedmeshes [Zander et al., 2015]. Figure 3.4 shows the concept of this approach.



Inactive edgesActive edgesActive facesInactive faces

a) 1D case b) 2D case

Figure 3.4: Multilevel hp meshes construction adopted from [Zander et al., 2015]

The linear independence is guaranteed by the introduction of the high-order shape functionsonly on the leaf elements. It is depicted in Figure 3.4 that the high-order shape functionsspan only over the elements, which do not have further refinement. Moreover, the lower levelelements include only linear modes in order to ensure the linear independence.

The compatibility condition is ensured in the same manner as for multiscale hp−d method bythe application of homogeneous Dirichlet boundary conditions on the overlay meshes [Zanderet al., 2015].

23

Chapter 4

The multiscale hp-d method forheat transfer problems

In this chapter, the idea of the multiscale hp − d method is applied to the heat transferanalysis. For full understanding of the conceptual and implementational aspects, this methodis developed for the simplified cases of one-dimensional heat equations: linear stationary,linear transient, nonlinear stationary and nonlinear transient. First, the idea is applied tothe governing equations, then implementational aspects are discussed with possible numericalmethods to be applied and, finally, the implementation results are verified. The multi-level hp−method is implemented in the in-house high-order FEM code AdhoC++. Theimplementation includes the monolithic evaluation of, for example, stiffness matrices throughall overlay meshes. In contrast, in the multiscale hp− d code, which is developed in Matlab,the partitioned approach is used. Therefore, the verification of the newly cerated code isperformed using already verified code in AdhoC++.

4.1 Linear stationary heat equation

In this section, the nonlinearity and transient effects of Equation 2.7 are neglected. Thenonlinear form A (T, k(T ), v) becomes linear B(T, v) and the inner product 〈ρcT , v〉 is set tozero.

4.1.1 Overlay approximation

Applying the hp− d approximation to the Equation 2.7 yields:

B(Tb + To, vb + vo) = F (vb + vo) ∀vb ∈ Vb ∧ ∀vo ∈ Vo, (4.1)

where the base and overlay spaces are linearly independent V = Vb⊕Vo. Therefore, Equation4.1 can be split into two equations:

24 4. The multiscale hp-d method for heat transfer problems

B(Tb + To, vb) = F (vb) ∀vb ∈ Vb

B(Tb + To, vo) = F (vo) ∀vo ∈ Vo.(4.2)

The bilinear form B(·, ·) can be split and brought to the right-hand side:

B(Tb, vb) = F (vb)−B(To, vb) ∀vb ∈ Vb

B(Tb, vo) = F (vo)−B(Tb, vo) ∀vo ∈ Vo.(4.3)

In matrix notation, Equation 4.2 becomes:[Kbb Kbo

KTbo Koo

] [Tb

To

]=

[Fb

Fo

]. (4.4)

The system of Equation 4.4 can be solved using Gauss-Seidel iterations. The coupled stiffnessmatrix is split into the lower and upper triangular matrix:

[Kbb 0KTbo Koo

] [Ti+1b

Ti+1o

]=

[Fb

Fo

]−[0 Kbo

0 0

] [Tib

Tio

], (4.5)

where i corresponds to the current Gauss-Seidel iteration. Equation 4.5 yields:

KbbTi+1b = Fb −KboT

io

KooTi+1o = Fo −KT

boTi+1b .

(4.6)

The coupling terms can be considered as an extra source term, originating from the fluxesof the overlay or the base mesh. The extra-flux term in Equation 4.6 can be represented asfollows:

KboTio =

∫Ωo

BTb k(x)BodΩoT

io =

∫Ωo

BTb k(x)εiodΩo

KTboT

i+1b =

∫Ωo

BTo k(x)BbdΩoT

i+1b =

∫Ωo

BTo k(x)εibdΩo,

(4.7)

where Ωo is the overlay domain.

4.1.2 Implementational aspects

Algorithm 1 shows the structure of the implemented hp−d method for stationary linear heatproblems [Krause and Rank, 2003].

Coupling terms in Equation 4.7 are computed with the help of composed Gauss quadrature.The numerical integration is performed on the normalized domain [−1, 1], therefore, all quan-tities in physical space have to be mapped to the parameter space ξ. Considering this, the

4.1. Linear stationary heat equation 25

Algorithm 1 : hp-d algorithm for stationary heat problem with linear elements

1: Create base mesh2: Set ansatz order p = 1 on base mesh3: Create overlay mesh in specified region of interest with n elements4: Set Gauss-Seidel iteration counter i = 05: Repeat:6: Compute base solution:7: Compute stiffness matrix of base problem Kbb

8: if i = 0 then9: Set coupling term as load increment Fi−1

bo = 010: else11: Compute coupling term as load increment Fi−1

bo =∫

Ωo

BTb kBodΩoT

i−1o

12: end if13: Solve KbbTi

b = Fb − Fi−1bo

14: Compute overlay solution:15: Compute stiffness matrix of overlay problem Koo

16: Compute coupling term as load increment Fiob =

∫Ωo

BTo kBbdΩoT

ib

17: Solve KooTio = Fo − Fi

ob

18: Compute strain energy U (T ) = 12 (B(Tb, Tb) + 2B(To, Tb) + B(To, To))

19: Check convergence of strain energy of complete solution20: i = i+ 121: Until converged or maximum number of iterations is reached22: Compute complete solution

coupling terms can be rewritten as follows:

KboTio =

∫Ωo

BTb k(x)BodΩoT

io =

1∫−1

∂NTb

∂ξb

∂ξb∂x

k(x)∂No

∂ξo

∂ξo∂x‖J‖dξoTi

o, (4.8)

KTboT

i+1b =

∫Ωo

BTo k(x)BbdΩoT

i+1b =

1∫−1

∂NTo

∂ξo

∂ξo∂x

k(x)∂Nb

∂ξb

∂ξb∂x‖J‖dξoTi+1

b , (4.9)

where ξ is the element natural coordinate.

As mentioned above, the coupling stiffness can be considered as an extra source term. Astraight forward distribution of the integration points on the base mesh would lead to inaccu-rate results due to under-integration of the mapped discontinuous overlay fluxes distributionalong the base mesh element. In order to compute the integral accurately, the integrationdomain must be subdivided according to the inner-element boundaries. Gauss points are dis-tributed along the overlay mesh and projected to the base mesh, where respective derivativesof shape functions are evaluated (see Figure 4.1).

Having a closer look at the terms in Equation 4.8 and 4.9, it can be noticed that in case oflinear shape functions on both base and overlay mesh and a constant material distribution



Figure 4.1: Integration on the constructed base and overlay domains

along the element, these coupling terms disappear and the system in Equation 4.4 becomescompletely decoupled [Krauser, 1996].

In order to prove this, one base element, which is overlayed by two linear elements, is consid-ered. The shape function are represented by Equation 4.10 in the overlay parameter spaceand the base parameter space.

N1o =

1

2(1− ξo) N1

b =1

2(1− ξb)

N2o =

1

2(1 + ξo) N2

b =1

2(1 + ξb) .

(4.10)

Now, considering a constant thermal conductivity k, Equation 4.8 is expanded as follows:

KboTio =

1∫−1

[−1

2

1

2] · 2

hb·k ·

[−1

2

12

]· 2

ho· ho

2dξo ·Ti

o = k · 2

hb

1∫−1

(−1

4+

1

4

)dξo ·Ti

o = 0. (4.11)

The same holds for Equation 4.9:

KTboT

i+1b =

1∫−1

[−1

2

1

2] · 2

ho· k ·

[−1

2

12

]· 2

hb· ho

2dξo ·Ti+1

b = k · 2

hb

1∫−1

(−1

4+

1

4

)dξo ·Ti+1

b = 0.

(4.12)

The stiffness matrix of base elements must be computed applying composed Gauss integrationdue to its C0 continuity within the element.

To sum up, the four following groups of computations should be considered separately:

• Stiffness matrices of base elements Kbb in Ωb \ Ωo, which do not have hierarchicalrefinements, are computed with standard Gauss integration

• Stiffness matrices of base elements Kbb in Ωo are computed by composed integration


• Coupling terms for KTboT

i+1b and KboT

io are computed by composed integration, dis-

tributing Gauss points along the overlay elements

• The stiffness matrix of the overlay elements Koo is computed applying standard Gaussintegration on the domain Ωo

The same holds for the calculation of the force vector. Within the refinement zone, the forcevector F should be computed using composed integration.

In the post processing step the strain energy of the system cannot be simply computed bythe sum of the strain energies of the overlay and base solutions, because B is a bilinear form[Krause and Rank, 2003]. Equation 4.13 shows the derivation of the strain energy formula.

U (T ) =1

2B(T, T ) =

1

2(B(Tb, Tb + To) + B(To, Tb + To)) =

=1

2(B(Tb, Tb) + 2B(To, Tb) + B(To, To)) . (4.13)

Then, the energy norm, as the measure of the temperature solution, can be used as theconvergence criteria for the Gauss-Seidel iterations:

‖T‖E(Ω) =√

U (T ) (4.14)

4.1.3 Verification

In order to verify the implemented multiscale hp − d code, the results were compared withmultilevel computations first, which are implemented and verified in AdhoC+ +, and secondwherever possible with an analytical solution. All verifications were performed with increasingcomplexity:

• Constant material distribution and sinusoidal load

• Linear material distribution and constant load

• Sinusoidal material distribution and sinusoidal load

The different setups are summarized in Table 4.1. For the considered test cases f(x) representsthe applied source term, m(x) the material distribution, Dirichlet BC indicates that both endsof the considered bar are fixed and Neumann BC that for the right end of the bar homogeneousNeumann boundary condition is applied.

For each case, a full overlay Ωo and a partial overlay were considered. In case of a partialoverlay, each base element was overlayed with 2 elements. The solutions were compared withthe multilevel hp solution as follows:

e =‖Thp−multilevel − Thp−d‖E(Ω)

‖Thp−multilevel‖E(Ω). (4.15)


Moreover, for each test case the number of Gauss-Seidel iterations until convergence isachieved are shown.

Test case Configuration Full overlay Partial overlay

1f(x) = −sin(8x)

k(x) = 1Neumann BC

Ωo = [0.0, 1.0] Ωo = [0.1, 0.3]e = 4.6459 · 10−29 e = 1.7413 · 10−29

5 iterations 1 iteration

2f(x) = −5

k(x) = 5x+ 10Dirichlet BC

Ωo = [0.0, 1.0] Ωo = [0.3, 0.6]e = 1.0067 · 10−30 e = 4.1245 · 10−31

5 iterations 5 iterations

3f(x) = −sin(8x)k(x) = sin(x) + 10

Neumann BC

Ωo = [0.0, 1.0] Ωo = [0.1, 0.3]e = 1.0462 · 10−29 e = 2.0610 · 10−30

4 iterations 4 iterations

Table 4.1: Verification of the multiscale code with the multilevel hp-code

Test case 1: Constant material distribution and sinusoidal load

For the first test case, unit constant thermal conductivity and sinusoidal distributed loadwere considered:

− ∂

∂x

(∂T

∂x

)= −sin(8x) in x ∈ [0, 1] (4.16)

T (0) = 0.0 at ΓD (4.17)

−∂T∂x

= 0 at ΓN (4.18)

For this case an analytical solution is known:

T (x) = − 1

64sin(8x) +

cos(8)

8x (4.19)

The Gauss-Seidel iterations converge after 1 iteration due to the orthogonality of the couplingterm. This was shown before in Equations 4.11 and 4.12.

Figure 4.2 shows the base mesh solution and the total overlay mesh solution using 10 baseelements and an overlay mesh in the interval of x ∈ [0.1, 0.3]. The overlay mesh contains 5elements per base element. A comparison with the analytical solution is shown in Figure 4.3.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−3

−2

−1

0

1·10−2

X coordinate

Tem

per

atu

re,

[oC

]

Base mesh solution Overlay mesh solution

Figure 4.2: Temperature distribution for the constant material distribution and sinusoidal load

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−3

−2

−1

0

1·10−2

X coordinate

Tem

per

atu

re,

[oC

]

Total solution Analytical solution

Figure 4.3: Total temperature distribution for the constant material distribution and sinusoidal loadcompared to analytical solution


Test case 2: Linear material distribution and constant load

The differential equation of the test case 2 reads:

− ∂

∂x

((5x+ 10) · ∂T

∂x

)= −5 in x ∈ [0, 1] (4.20)

T (0) = 0.0 at ΓD (4.21)

T (1) = 0.0 at ΓD (4.22)

For 10 base elements and an overlay mesh in the interval of x ∈ [0.3, 0.6] with 5 overlayelements per base element the total and total overlay solution are shown in Figure 4.4.

For this case, the analytical solution is plotted in Figure 4.5. In total 5 Gauss-Seidel itera-tions are needed for the strain energy change not to exceed 10−15 (see Figure 4.6). In thistest case the thermal diffusivity is dependent on the spatial distribution, and therefore theorthogonality of the coupling term does not hold.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5

−4

−3

−2

−1

0·10−2

X coordinate

Tem

per

atu

re,

[oC

]


Figure 4.4: Temperature distribution for the linear material distribution and constant load


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−5

−4

−3

−2

−1

0·10−2

X coordinate

Tem

per

atu

re,

[oC

]


Figure 4.5: Total temperature distribution for the linear material distribution and constant loadcompared to analytical solution

2 3 4 5

10−16

10−14

10−12

10−10

10−8

10−6

Gauss-Seidel iterations

Err

or

inth

een

ergy

nor

m,

[-]

Figure 4.6: Error in the energy norm for the linear material distribution and constant load


Test case 3: Sinusoidal material distribution and sinusoidal load

In the following, test case 3 is considered:

− ∂

∂x

((sin(x) + 10) · ∂T

∂x

)= −sin(8x) in x ∈ [0, 1] (4.23)

T (0) = 0.0 at ΓD (4.24)

−(sin(x) + 10)∂T

∂x

∣∣∣∣x=1

= 0 at ΓN. (4.25)

For this case, the analytical solution is presented in Figure 4.8. The Gauss-Seidel schemeconverges after 4 iterations, when the strain energy relative increment does not exceed 10−15

(see Figure 4.9).

The same setup of ten base elements with the overlay zone of x ∈ [0.1, 0.3] is chosen. Thebase solution with the total overlay solution is presented in Figure 4.7.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−3

−2

−1

0

·10−3

X coordinate

Tem

per

atu

re,

[oC

]


Figure 4.7: Temperature distribution for the sinusoidal material distribution and sinusoidal load


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−3

−2

−1

0

·10−3

X coordinate

Tem

per

atu

re,

[oC

]


Figure 4.8: Total temperature distribution for the sinusoidal material distribution and sinusoidalload compared to analytical solution

2 3 4

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5


Err

orin

the

ener

gy

nor

m,

[-]

Figure 4.9: Error in the energy norm for sinusoidal material distribution and sinusoidal load


4.1.4 Significance of the coupling terms

For implementation purposes, the influence of the coupling term on the computation is stud-ied. The coupling term, originating from the effect of the overlay correction on the base meshKboTo, influences the base force vector Fb. The coupling term KobTb acts as the extra sourceterm to the right-hand side of the overlay mesh Fo. Therefore, the relative significance iscompared. As only the magnitudes of two vectors can be linked to find the relative influence,the L2-norm is used.

First, the L2-norm of each vector is calculated after constraining the system. Then the overlaysignificance So and base significance Sb are computed as follows:

So =‖KboTo‖L2

‖Fb‖L2

Sb =‖KobTb‖L2

‖Fo‖L2

.

(4.26)

Test case 1 was discussed above and represents the special case of no coupling. It wasproven, that if any of the coupling terms are canceled, the numerical solution converges tothe analytical one. Therefore, this case is of no interest and test case 3 is considered.

Influence of the spatial refinement on the coupling terms for test case 3

First, the coupling terms significance is assessed with respect to the spatial refinement. Fortest case 3, the same settings as in Table 4.1 are used. For the full overlay the number ofoverlay elements per base element is increased from 1 to 150. For the partial overlay the sameis performed only within the zone of x ∈ [0.1; 0.3]. The results are summarized in Table A.3.It becomes apparent, that the base and the overlay coupling term has constant influence,independent on the performed mesh refinement.

Influence of neglecting the coupling terms on the Gauss-Seidel convergence fortest case 3

For further investigation, the specific setting with 20 overlay elements per base element (fulloverlay) is considered.

First, full coupling is performed (So 6= 0;Sb 6= 0 in Table 4.3), then, the coupling term KboTo

is set to zero (So = 0 in Table 4.3), last, KobTb is canceled (Sb = 0 in Table 4.3).

The overlay significance So and the base significance Sb are recorded within each Gauss-Seideliteration. The total hp−d solution after the convergence of Gauss-Seidel iterations is reached,is compared to the analytical one. The final L2-error is indicated for each of the couplingcases in Table 4.3.

When the overlay coupling term is not considered, the Gauss-Seidel method is convergedafter two iterations. Disregarding the base coupling term yields three Gauss-Seidel iterationsneeded to converge. Table 4.3 shows that in the case of non-complete coupling the Gauss-Seidel procedure converges faster. However, it is clear, that the quality of the convergedsolution drops.


Condition GS 1 GS 2 GS 3 GS 4 GS 5 L2 error

So 6= 0 0 2.94 · 10−7 2.94 · 10−7 2.94 · 10−7 2.94 · 10−7

1.03 · 10−8

Sb 6= 0 1.05 · 10−4 1.05 · 10−4 1.05 · 10−4 1.05 · 10−4 1.05 · 10−4

So = 0 0 0 - - -5.85 · 10−7

Sb 6= 0 1.05 · 10−4 1.05 · 10−4 - - -

So 6= 0 0 2.95 · 10−7 2.95 · 10−7 - -2.00 · 10−7

Sb = 0 0 0 0 - -Table 4.3: Influence of the neglection of the coupling terms for Test Case 3

Influence of neglecting the coupling terms on the spatial convergence for testcase 3

The results in Table 4.3 demonstrate that neglecting the coupling terms, the Gauss-Seideliterations converge faster. However, it is necessary to evaluate, when any of the coupling termis neglected, while increasing of the number of the overlay elements, the numerical solutionwould converge to the analytical one.

Table A.4 summarizes the convergence studies, neglecting one or another coupling term. Itis clear, that none of them can be neglected in this case, as the numerical solution convergesto a wrong value (See Figure 4.10).

20 40 60 80 100 120 14010−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Number of the overlay elements per each base element

L2

nor

mof

the

nu

mer

ical

solu

tion

,[

]

So 6= 0, Sb 6= 0 So 6= 0, Sb = 0 So = 0, Sb 6= 0

Figure 4.10: Influence of the neglecting of the coupling terms on the convergence behavior for testcase 3

The same tests were performed for the test case 2. The results can be found in Tables A.1 -A.2. In the same manner as for test case 3, neglecting one or another coupling terms leadsto a non-converged solution.


4.2 Linear transient heat equation

In this section, the transient effect is taken into consideration, while keeping the materialdistribution linear. Therefore, the nonlinear form in Equation 2.7 is again simplified to abilinear.


Applying the concept of the hp− d approximation, described in Section 3.2, the weak formu-lation of the transient problem (Equation 2.7) yields:

〈ρc(Tb + To), vb + vo〉+ B(Tb + To, vb + vo) = F (vb + vo) ∀vb ∈ Vb ∀vo ∈ Vo. (4.27)

Equation 4.27 can be rewritten as follows:

〈ρc(Tb + To), vb〉+ B(Tb + To, vb) = F (vb) ∀vb ∈ Vb

〈ρc(Tb + To), vo〉+ B(Tb + To, vo) = F (vo) ∀vo ∈ Vo.(4.28)

In the matrix notation Equation 4.28 results in:[Mbb Mbo

MTbo Moo

] [Tb

To

]+

[Kbb Kbo

KTbo Koo

] [Tb

To

]=

[Fb

Fo

]. (4.29)

Applying the Backward-Euler time integration scheme (see Equation 2.34), Equation 4.29becomes:

[Mbb + ∆tKbb Mbo + ∆tKbo

MTbo + ∆tKT

bo Moo + ∆tKoo

] [∆Tb

∆To

]= ∆t

([Fb

Fo

]−[Kbb Kbo

KTbo Koo

] [Tnb

Tno

]), (4.30)

where ∆T corresponds to the increment of the temperature solution, Tn is the temperaturesolution of the previous time step and Fb and Fb corresponds to the source term, evaluatedat the current time step n+ 1.

After solving the coupled system of Equations 4.30, T is updated for the current time stepaccording to Equation 2.35.

The system of Equations 4.30 is solved using the Gauss-Seidel scheme in order to decouplebase and overlay problems:

4.2. Linear transient heat equation 37

[Mbb + ∆tKbb 0MT

bo + ∆tKTbo Moo + ∆tKoo

] [∆Ti+1

b

∆Ti+1o

]=

= ∆t

[Fb −KbbT

nb −KboT

no

Fo −KTboT

nb −KooT

no

]−[0 Mbo + ∆tKbo

0 0

] [∆Ti

b

∆Tio,

](4.31)

where i corresponds to the current Gauss-Seidel iteration. Equation 4.31 can be rewritten astwo separate equations:

(Mbb + ∆tKbb) ∆Ti+1b = ∆t (Fb −KbbT

nb −KboT

no )− (Mbo + ∆tKbo) ∆Ti

o

(Moo + ∆tKoo) ∆Ti+1o = ∆t

(Fo −KT

boTnb −KooT

no

)−(MT

bo + ∆tKTbo

)∆Ti+1

b .(4.32)

Equation 4.32 can be reformulated to isolate the coupling terms for the force vector as follows:

(Mbb + ∆tKbb) ∆Ti+1b = ∆t (Fb −KbbT

nb )−Mbo∆Ti

o −∆tKbo

(Tno + ∆Ti

o

)(Moo + ∆tKoo) ∆Ti+1

o = ∆t (Fo −KooTno )−MT

bo∆Ti+1b −∆tKT

bo

(Tnb + ∆Ti+1

b

).

(4.33)

The coupling terms are considered in the similar manner as in Section 4.1.1. To this end, theextra-source term in Equation 4.33 is reformulated as follows:

Mbo∆Tio + ∆tKbo

(Tno + ∆Ti

o

)=

=

∫Ωo

NTb ρcNodΩo∆Ti

o + ∆t

∫Ωo

BTb kBodΩo

(Tno + ∆Ti

o

)MT

bo∆Ti+1b + ∆tKT

bo

(Tnb + ∆Ti+1

b

)=

=

∫Ωo

NTo ρcNbdΩo∆Ti+1

b + ∆t

∫Ωo

BTo kBbdΩo

(Tnb + ∆Ti+1

b

).

(4.34)

4.2.2 Time discretization of the overlay approximation

The overlay approximation allows for the use of a sub stepping scheme, i.e. a smaller timediscretization of the overlay approximation than of the base mesh (see Figure 4.11). To thisend, Equation 4.29 is rewritten as follows, using a Backward-Euler time integration scheme:


tb

to

t0b t1b t2b tendb

t0o t1o t2o t3o t4o t8o tendo

Figure 4.11: Time sub stepping within multiscale hp− d approach

[Mbb Mbo

MTbo Moo

] [∆Tb∆tb∆To∆to

]+

[Kbb Kbo

KTbo Koo

] [Tnbb

Tnoo

]+

[Kbb Kbo

KTbo Koo

] [∆Tb

∆To

]=

[Fb

Fo

](4.35)

where ∆tb is the time step of the base mesh, ∆to the time step of the overlay mesh, Tnbb

the previous time step solution on the base mesh, and Tnoo the previous time step solution

on the overlay mesh. It is necessary to mention, that the coupling steps t4o and t1b should besynchronized in time, i.e. corresponds to the same time state.

Rewriting Equation 4.35 yields:

[1

∆tbMbb + Kbb

1∆to

Mbo + Kbo1

∆tbMT

bo + KTbo

1∆to

Moo + Koo

][∆Tb

∆To

]=

[Fb −KbbT

nbb −KboT

noo

Fo −KTboT

nbb −KooT

noo

](4.36)

Using Gauss-Seidel iterations, Equation 4.36 is decoupled as follows:

(1

∆tbMbb + Kbb

)∆Ti+1

b =(Fb −KbbT

nbb −KboT

noo

)−(

1

∆toMbo + Kbo

)∆Ti

o(1

∆toMoo + Koo

)∆Ti+1

o =(Fo −KT

boTnbb −KooT

noo

)−(

1

∆tbMT

bo + KTbo

)∆Ti+1

b

(4.37)

For convenience of implementation Equation 4.37 is rewritten:

(1

∆tbMbb + Kbb

)∆Ti+1

b =(Fb −KbbT

nbb

)−Kbo

(Tnoo + ∆Ti

o

)− 1

∆toMbo∆Ti

o(1

∆toMoo + Koo

)∆Ti+1

o = (Fo −KooTnoo )−KT

bo

(Tnbb + ∆Ti+1

b

)− 1

∆tbMT

bo∆Ti+1b

(4.38)

There are two possible ways of implementing Equation 4.38. The difference lies in the ap-proximation of the solution betwen two base time steps tib and ti+1

b :


• The solution on the base mesh Tnbb can be considered to be constant, while the overlay

solution is changing within smaller overlay time steps (From now on referred to asMethod 1)

• The sub stepping can be performed on the complete hp-d solution Thp−d = T0 + Tbwithin the overlay domain Ωo, which allows the base solution to be “corrected“ withinthe overlay time steps (From now on referred to as Method 2).

The first option is computationally less expensive because it constrains the overlay solutionto oscillate around a fixed base solution within one base time step. This constraint does notexist in the second option. The aim of further sections is to investigate, how this approachinfluences the quality of the final solution. For better understanding, Figure 4.13 shows anexample of the development of the solution within one base time step. At the beginning, abase solution T 0

b (in blue) and an overlay solution T 0o (in red) at time step t0 = t0o = t0b are

considered. After time step t0, four sub steps are introduced.

In Method 1, the system is solved around the oscillating overlay part. After calculation of3 extra sub time steps, the overlay solution is developed as it is shown in Figure 4.12 ingreen. It is clear, that the solution at the nodes 1 − 3 and 7 − 8 remains unchanged withinconsidered sub time steps. But also, the solution at the node 5 will remain unchanged withinsub stepping. It is clear, that the base solution is not affected at all, while performing thesub stepping on the oscillating overlay part (marked in blue in Figure 4.12).

If Method 2 is applied for the calculation of the sub steps, the complete solution is takenThp−d = To + Tb into account. First, the overlay solution is projected onto the base meshsolution space and for further calculations the complete hp− d solution is taken (marked inred on the bottom left Figure 4.13). After three sub steps the solution is developed as it isshown in the bottom right Figure 4.13. For the coupling time step t4o = t1b , the separation oftwo scales is necessary again. It is clear, that this approach allows the degree of freedom atnode 5 to change within each considered sub step.

This observation must be considered if the base solution is strongly changing on the basedomain within sub time steps t0o − t3o for Method 1. Moreover, if the overlay mesh coversa large region of interest, as an overlay solution is introduced to capture the local stronggradients of the base solution, fixing the overlayed base-mesh nodes will decrease the qualityof the solution.


12

34

6

7 8

5

12

3

4

6

7 85

t3o

t0 t3

12

34

6

7 8

5t0o

12

3

4

6

7 85

Figure 4.12: Development of hp− d solution within one base time step using Method 1

12

34

56 7 8

12

34

6

7 8

5

12

3

4

6

7 85

12

3

4

6

7 85

t3o

t0

t0o

t3

Figure 4.13: Development of hp− d solution within one base time step using Method 2

It is important to note, that time step ∆tb in Equation 4.38 is different for refined and non-refined elements. As the solution at the non-refined elements, which are not changing withingsub time steps ∆tb, is considered. For the overlayed base elements it is necessary to consider∆to instead.


4.2.3 Implementation of hp− d method

Algorithm 2 shows the structure of the implemented hp− d method for transient linear heatproblems.

Algorithm 2 : hp-d algorithm for transient heat problems with linear elements

1: Create base mesh2: Set Tnb

b and Tnoo to initial condition

3: for iT imeStep = 1 .. numberOfT imeSteps do4: Create/Move overlay mesh5: Set ∆T ib and T io to zero6: Set current time7: if Substepping = true8: for iT imeStep = 1 .. numberOfSubT imeSteps do9: Create overlay transient problem

10: Solve Kdyn∆Tno+1o = ∆to (Fo −KooT

noo )

11: Prepare solution for the coupling12: end for13: end if14: for iGaussSeidel = 1 ..maximumNumberOfIterations do15: Compute base solution increment:

16: Compute Kdyn =(

1∆tb

Mbb + Kbb

)17: if i = 0 then18: Set Fi

bo = 019: else20: Compute Fi

bo = 1∆to

Mbo∆Tio + Kbo

(Tnoo + ∆Ti

o

)21: end if22: Solve Kdyn∆Ti+1

b =(Fb −KbbT

nbb

)− Fi

bo

23: Compute overlay solution increment:

24: Compute Kdyn =(

1∆to

Moo + Koo

)25: Compute Fi+1

ob = 1∆tb

MTbo∆Ti+1

b + KTbo

(Tnbb + ∆Ti+1

b

)26: Solve Kdyn∆Ti+1

o = (Fo −KooTnoo )− Fi+1

ob

27: Check convergence of L2-norm of complete solution28: if L2-norm is converged29: break Gauss-Seidel Iterations30: end if31: end for32: Increment base mesh solution Tnb+1

b = Tnbb + ∆Ti+1

b

33: Increment overlay mesh solution Tno+1o = Tno

o + ∆Ti+1o

34: end for35: Compute complete solution

For linear transient problems, the mesh is created newly in each time step, as the movementof the mesh is allowed. The coupling terms in Equation 4.34, the stiffness and mass matricesof the base elements, which have a hierarchical overlay, are computed with the help of Gaussquadrature in the same manner as explained in Section 4.1.2.

Special care should be taken for the application of boundary conditions for both cases, where


an L2-projection should be applied.

At first, the L2-projection of the initial condition on the complete finite element space, in-cluding the first required refinement (see Equation 2.25), is determined. For example, thep-rojection of f(x) = sin(8x) onto the base mesh of 10 elements together with the refined zonex ∈ [0.1, 0.5] with 20 elements is shown in Figure 4.14. To obtain the initial condition for theoverlay mesh, first an L2 projection on the complete domain is performed. The complete do-main includes all refinement levels, but considers the shape function space as a simply refinedone. This method is valid in the one-dimensional case, but needs to be reconsidered for thehigh-order modes and two-dimensions. Moreover, the shape functions from both meshes arenot available at this calculations moments. Therefore, other methods need to be developed.In the second step the solution of the base mesh is evaluated at the position of the overlaynodes. Then the corresponding degrees of freedom of the overlay mesh are subtracted by thisvalue (the difference between red and black lines in Figure 4.14).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

X coordinate

Yco

ord

inat

e

Projection on the base mesh Projection on the overlay meshInitial condition function

Figure 4.14: Computation of the initial condition for the base and overlay meshes

In the post processing step, the L2-norm of the complete solution is computed as follows:

‖T‖L2 =

∫Ω

T 2(x)dΩ. (4.39)

For the discrete solution the integral is computed numerically. As an error measure therelative error in the L2 norm is used:

‖e‖L2 =‖Tref − Thp−d‖L2

‖Tref‖L2

(4.40)


The work flow for the implementation of the sub stepping is as follows (see Figure 4.15):

• First, the multiscale hp − d problem is solved with the newly created overlay mesh attime step t0o = t0b

• Then, a new transient problem is created, based on the overlay mesh coordinates. Theinitial condition for the first time sub step t1o is computed from the hp−d solution of thecoupling step t0o = t0b . As already mentioned, the total solution is now projected ontothe overlay mesh (forward projection (FP) in Figure 4.15). Nonhomogenous DirichletBoundary conditions are then applied to the overlay problem, constraining the nodalvalues at the boundary of the overlay domain to the base mesh solution of the previouscoupled step t0o = t0b

• The overlay transient problem is solved as a monolithic transient problem within timesteps t1o,t

2o ... t

no

• Then the initial condition for the coupling step tno = t1b is computed projecting the hp−dsolution on the base and overlay domain (backward projection (BP) in Figure 4.15 ).Considering that the direct descendant nodes of the overlay mesh should be set to zero,the base nodes are assigned to the complete solution obtained from the previous timesub step. Then the difference between the hp − d solution and the projected overlaysolution is computed and applied to the overlay nodes.

• Finally, the multiscale hp − d solution is computed for the next coupling time stept4o = t1b

T 0o

T 0b

...T 0hp−d Tnhp−d

Tno

T 1b

FP BPsubstepsn

Figure 4.15: Workflow for sub stepping within one base time step.

This procedure is repeated for each set of time sub steps of the overlay mesh. The numberof considered overlay time steps can be varied.

4.2.4 Verification

Comparison with multilevel hp− solution, coupling at each time step

The verification of the multiscale hp− d code has been performed in the same manner as inSection 4. Test cases were held increasing complexity of the problem (see Table 4.5).

In Table 4.5, f(x, t) represents the applied heat source, m(x, t) the material distribution andT (x, 0) initial condition. For the applied flux the following equation was considered:

f(x, t) =1

0.02 ·√

2π

(x+ 0.8− t

0.022− (−x+ 0.2 + t)2

0.024

)exp

(−(x− 0.2− t)2

2 · 0.022

). (4.41)


This formula has been chosen for further verification as the analytical solution is known:

T (x, t) =1

0.02 ·√

2πexp

(−(x− 0.2− t)2

2 · 0.022

). (4.42)

The initial condition for test cases 3 and 4 are then formulated as follows:

T (x, 0) =1

0.02 ·√

2πexp

(−(x− 0.2)2

2 · 0.022

). (4.43)

For all test cases, 40 base elements were created to capture the physical effect with sufficientaccuracy. Simulations were run for t ∈ [0, 0.1] with 100 time steps. The overlay mesh wasmoved after half of the simulation time. For test case 3 and 4, the movement of the meshwas following, the movement of the Gaussian bell of the temperature distribution.

Case Configuration Full overlay Partial overlay Moving mesh

1f(x, t) = −sin(8x) · t

k(x, t) = 1T (x, 0) = 0.0

Ωo = [0.0, 1.0] Ωo = [0.55, 0.85] Ωo =[0.15, 0.45]→

[0.55, 0.85]e = 5.0361 · 10−29 e = 2.2479 · 10−29 e = 1.3080 · 10−29

6 iterations 5 iterations 5 iterations

2f(x, t) = −sin(8x) · tk(x, t) = sin(x) + 10

T (x, 0) = 0.0

Ωo = [0.0, 1.0] Ωo = [0.55, 0.85] Ωo =[0.15, 0.45]→

[0.55, 0.85]e = 4.0553 · 10−30 e = 2.0339 · 10−29 e = 2.8368 · 10−29


3f(x, t)-Gaussian bell

k(x, t) = 1T (x, 0)-Gaussian bell

Ωo = [0.0, 1.0] Ωo = [0.1, 0.4] Ωo =[0.05, 0.35]→

[0.15, 0.45]e = 7.3305 · 10−25 e = 2.1769 · 10−21 e = 1.1203 · 10−25


4f(x, t)-Gaussian bellk(x, t) = sin(x) + 10T (x, 0)-Gaussian bell

Ωo = [0.0, 1.0] Ωo = [0.1, 0.4] Ωo =[0.05, 0.35]→

[0.15, 0.45]e = 8.9374 · 10−28 e = 5.9423 · 10−25 e = 1.0936 · 10−27

6 iterations 5 iterations 5 iterationsTable 4.5: Verification of the multiscale code with the multilevel hp-code for transient test cases

Comparison with multilevel hp− solution, introducing sub stepping

In order to verify the sub stepping, only test case 3 from Table 4.5 is considered. The overlayis introduced in the spatial ranges as it is stated in the table. It was mentioned that thebase solution from one base time step to another is enforced to not to change strongly withinnon-overlayed zone, therefore test cases 1 and 2 are not optimal for verification.

First, a comparison of the multilevel hp-solution with multiscale approach is considered,constantly increasing the base time step. For each test case, a full (each element is overlayed


with two elements), a partial (each element within fixed region is overlayed with two elements)and a moving (the region of overlay is moved after half of total simulation time) overlays aretested. First, the reference solution of time step ∆t = 0.001 s is generated, coupling themultiscale hp − d solution after each time step. Now, the sub stepping is introduced. Thetime step of the overlay mesh ∆to = 0.001 s is kept constant for all simulations, while the timestep of the base mesh is constantly increased from ∆tb = 0.001 s to ∆tb = 0.05 s. The erroris computed using Equation 4.40. The reference solution Tref is the multi-level hp−solutiongenerated with the same configuration and a time step ∆t = 0.001 s.

Column “Method 1” (M1) in Table 4.6 shows the error introduced by increasing the of basetime step for the first approach to the sub stepping computation, namely solving the overlaysolution within sub time steps (see Section 4.2.2). It can be seen, that the error is constantlygrowing with increasing the base time step. This is due to the “fixation” of the base nodes,which have direct descendant overlay nodes, within sub time steps, shown in Figure 4.13 ingreen. In contrast, when the full hp − d solution is considered in the refined domain, theerror for the full domain overlay is constant (see column M2 Table 4.6). For the partial andmoving overlay with the the increase of base time step it can be observed that a constanterror is introduced. This error originates from the neglection of the change in the rest of thebase domain nodes from base time step tnb to tn+1

b (see nodes 1− 3 and 7− 8 in Figure 4.13).

The introduced error by the increase of the base mesh time step ∆tb for the full overlay isshown in Figure 4.16, for partial overlay in Figure 4.17 and for moving overlay in Figure 4.18.

∆tb, [s]L2 error, [ - ]

Full overlay Partial overlay Moving overlayM1 M2 M1 M2 M1 M2

0.001 7.33 · 10−25 7.33 · 10−25 2.18 · 10−21 2.18 · 10−21 1.12 · 10−25 1.12 · 10−25

0.002 2.03 · 10−7 1.36 · 10−26 2.06 · 10−7 3.09 · 10−10 2.06 · 10−7 3.06 · 10−11

0.004 1.79 · 10−6 4.46 · 10−27 1.84 · 10−6 1.94 · 10−9 1.84 · 10−6 1.61 · 10−9

0.005 3.16 · 10−6 3.29 · 10−27 3.25 · 10−6 2.22 · 10−9 3.25 · 10−6 3.07 · 10−9

0.01 1.56 · 10−5 1.78 · 10−27 1.59 · 10−5 1.29 · 10−9 1.59 · 10−5 5.77 · 10−9

0.02 6.67 · 10−5 1.54 · 10−27 6.69 · 10−5 3.88 · 10−10 6.69 · 10−5 2.05 · 10−9

0.025 1.04 · 10−4 2.02 · 10−27 1.04 · 10−4 1.37 · 10−9 1.04 · 10−4 1.13 · 10−9

0.05 3.79 · 10−4 3.60 · 10−27 3.79 · 10−4 4.72 · 10−9 3.79 · 10−4 4.01 · 10−9

Table 4.6: Error in the energy norm depending on the increase of the base time step ∆tb for test cases 2 and 3


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

·10−2

10−29

10−25

10−21

10−17

10−13

10−9

10−5

Time step of the base mesh ∆tb

L2

erro

r,[

-]

Full overlay method 1Full overlay method 2

Figure 4.16: Error introduced by the increase of the base time step ∆tb for a full overlay

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

·10−2

10−22

10−19

10−16

10−13

10−10

10−7

10−4


L2

erro

r,[

-]

Partial overlay method 1Partial overlay method 2

Figure 4.17: Error introduced by the increase of the base time step ∆tb for a partial overlay


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

·10−2

10−27

10−24

10−21

10−18

10−15

10−12

10−9

10−6


L2

erro

r,[

-]

Moving overlay method 1Moving overlay method 2

Figure 4.18: Error introduced by the increase of the base time step ∆tb for a moving overlay

Convergence studies

In order to investigate the convergence behavior of the multiscale solution in space, test case3 from Table 4.5 was considered. The overlay is introduced in the spatial ranges as it isstated in Table 4.5. In order to calculate the L2-error of the numerical solution (see Equation4.40), the analytical solution (see Equation 4.42) was used as reference. The L2-error wascomputed using Gauss quadrature.

First, the studies were performed fixing the discretization to 40 base elements with a fulloverlay of two elements per base element. Ten base time steps were considered, constantlyincreasing the number of sub steps per base time step. When the error levels off, the er-ror introduced by the spatial discretization is prevalent, therefore a spatial refinement isperformed.

Figure 4.19 shows a convergence graph for full and partial overlay. It is shown, that whenthe base nodes inside the overlay domain are considered unchanged from one base time stepto another, the solution does not converge (red and green lines in the graph). Therefore, nostudy with the spatial refinement will be performed for this method. It is concluded thatthis approach can be used only in case of the overlay of a single base element. For furtherconvergence studies of method 2, a spatial refinement is performed fixing the number of substeps to 300 per base time step. Figure 4.20 shows the convergence graph for this case.Moreover, it was compared whether the refinement of the base domain or the overlay domainfor the case of full overlay leads to a different error. For example, for the case with 40 baseelements and each of them overlayed with 4 elements and 80 base elements with 2 overlayedthe difference between L2-norms was 9.77 · 10−16. The results are summarized in Table B.1and B.2.


101 1020.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1·10−3

Number of overlay sub time steps per base time step

L2

erro

r,[

-]

Full overlay method 1Full overlay method 2

Partial overlay method 1Partial overlay method 2

Figure 4.19: Convergence studies on time refinement

2 4 6 8 10 12 14 16 18 20

10−7

10−6

10−5

10−4

Number of overlay elements per base element

L2

erro

r,[

-]

Full overlay method 2Partial overlay method 2

Figure 4.20: Convergence studies on space refinement


4.2.5 Significance of the coupling terms

The analysis of the significance of the coupling terms is performed in the same manner asfor the linear stationary cases. As there are two extra coupling terms now for each of themeshes, the influence of them is evaluated separately, using the following definitions:

SoK =‖Kbo

(Tno + ∆Ti

o

)‖L2

‖Fb‖L2

(4.44)

SoM =‖Mbo∆Ti

o‖L2

‖Fb‖L2

(4.45)

SbK =‖Kob

(Tnb + ∆Ti+1

b

)‖L2

‖Fo‖L2

(4.46)

SbM =‖Mob∆Ti+1

b ‖L2

‖Fo‖L2

(4.47)

First, test case 3 from Table 4.5 was considered. Th respective results are summarized inTables B.3 - B.5. If the thermal conductivity is kept constant, then the terms SoK andSbK can be neglected (see Table B.3). As soon as the thermal conductivity is dependent onthe spatial coordinate x (test case 4), the stiffness coupling terms can not be neglected (seeTable B.5). In contrast, the mass coupling terms were significant for both cases. As thesignificance of the coupling terms is similar for the partial overlay, this case is not consideredhere. The same holds for the cases when the spatial resolution is improved. For the tests 40base elements and ten base time steps are considered. The number of the overlay elementsper each base element is increased. Five time sub steps are taken into account. The resultscan be seen for the test case 3 in Table B.4 and for the test case 4 B.6.

Then for the test case 3 it was tested, how the width of the solution affects the significanceof the coupling terms. The width was reduced from 0.02 till 0.002 (see Figure 4.21). Inorder to capture such a small width, the spatial and temporal resolution has to be very fine.Therefore, for all test cases the base number of elements was kept as 200, each was overlayedby two elements. Twenty base time steps are considered with five sub steps per each. Theresults can be seen in Table B.7. It can be noticed, that the significance of the mass termon the overlay domain is increasing, while the mass coupling term for the base is loosing itsvalue.

Therefore, out of all performed tests the following conclusion can be made. None of thecoupling terms can be neglected, as they are highly dependent on the considered case. Forthe cases with high dependency of the material coefficients on the spatial coordinates, all fourterms are equally important to obtain a convergent solution.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

X coordinate

Tem

per

ature

fiel

d,

[]

σ = 0.02σ = 0.002

Figure 4.21: Influence of the width of the Gaussian bell on the solution field

4.3 Nonlinear stationary heat equation

In this section nonlinear stationary case is considered. In Equation 2.7, the time derivativeis assumed to be zero.


The overlay concepts can be also applied to nonlinear stationary problems. As the hierarchicaloverlay is a decomposition of the continuous space V :

V = Vb ⊕ Vo (4.48)

and the linear dependencies between base and overlay spaces are eliminated, the nonlinearform A (T, k(T ), v) is linear in the virtual temperature function v. Therefore, the nonlinearform can be split and Equation 2.7 will lead to the following system of equations:

A (T, k(T ), vb) = F (vb) ∀vb ∈ Vb (4.49)

A (T, k(T ), vo) = F (vo) ∀vo ∈ Vo. (4.50)

Inserting Equation 3.1 into Equation 4.50 yields:

4.3. Nonlinear stationary heat equation 51

A (To + Tb, k(To + Tb), vb) = F (vb) ∀vb ∈ Vb (4.51)

A (To + Tb, k(To + Tb), vo) = F (vo) ∀vo ∈ Vo. (4.52)

After discretization of Equation 4.52, the compact matrix notation of the nonlinear systemof equations is written:

[Kbb(To + Tb) Kbo(To + Tb)KTbo(To + Tb) Koo(To + Tb)

] [Tb

To

]=

[Fb

Fo

], (4.53)

where

Kbb(To + Tb) =

∫Ω

BTb k(x,NbTb + NoTo)Bb dΩ (4.54)

Kbo(To + Tb) =

∫Ω

BTb k(x,NbTb + NoTo)Bo dΩ (4.55)

Koo(To + Tb) =

∫Ω

BTo k(x,NbTb + NoTo)Bo dΩ, (4.56)

As the hp− d approximation is aiming to solve separately the nonlinear system of equations4.53, there are different approaches to achieve the decoupling of base and overlay equations.The whole range of the nonlinear iterative solvers can be found in the literature (see e.g.[Ortega and Rheinboldt, 2000]). The straightforward way would be the application of aNewton-Gauss-Seidel method. At first, the system of equations 4.53 is linearized and theniteratively solved using Gauss-Seidel iterations. Though this seems to be the most efficientway, for further introduction of sub stepping one would need to have the Gauss-Seidel iterationloop outside of the Newton iterations. For this reason, other methods as the m-step blockGauss-Seidel-Newton and the m-step block Jacobi-Newton, will be introduced.

Method 1: Newton-Gauss-Seidel approach

The idea of a Newton-Raphson iteration scheme can be used directly at the level of the partialdifferential equation (PDE), prior to the discretization (see [Langtangen, 1999]). This willlead to the replacement of the nonlinear PDE by a system of linear PDEs, which can bediscretized afterwards. This approach simplifies the following derivations. The starting pointis the nonlinear problem (as already stated in Equation 4.50):

Rb(T ) = F (vb)−A (T, k(T ), vb)!

= 0 ∀vb ∈ Vb (4.57)

Ro(T ) = F (vo)−A (T, k(T ), vo)!

= 0 ∀vo ∈ Vo. (4.58)

First, the nonlinear residual for both, the base and overlay spaces is linearized, using aTaylor-series expansion and omitting high-order terms:


R(T k) ≈ R(T k−1) +DR(T k−1)[∆T ], (4.59)

where k = 1...N is Newton iteration steps and DR(T k−1)[∆T ] is the Gateaux derivative ordirectional derivative. The temperature field at the next iteration step T k(x) is approximatedas:

T k(x) = T k−1(x) + ∆T (x). (4.60)

The residual R(T ) at any iteration step is required to be zero. Therefore, the linear PDE tobe solved in each iteration becomes:

DRb(Tk−1)[∆T ] = −Rb(T k−1) (4.61)

DRo(Tk−1)[∆T ] = −Ro(T k−1), (4.62)

where:

Rb(Tk−1) =

∫Ω

f(x)vbdΩ−∫Ω

v′bk(x, T k−1)(T k−1)′dΩ (4.63)

Ro(Tk−1) =

∫Ω

f(x)vodΩ−∫Ω

v′ok(x, T k−1)(T k−1)′dΩ. (4.64)

In order to complete the formulation of Equation 4.61, the directional derivative of the resid-ual, evaluated at the previous iteration step has to be determined. The definition of thedirectional derivative is given in [Bonet and Wood, 2008]:

DR(T )[∆T ] =

[d

dεR(T + ε∆T )

]ε=0

. (4.65)

The directional derivative has similar properties as the normal derivative. Applying theproduct rule, the directional derivative of the base residual can be expressed as:


DRb(Tk−1)[∆T ] =

∫Ω

:0D(f(x)vb)[∆T ]dΩ−

−∫Ω

:

0D(v′b)[∆T ]k(x, T k−1)(T k−1)′dΩ−

−∫Ω

v′bD(k(x, T k−1))[∆T ](T k−1)′dΩ−

−∫Ω

v′b k(x, T k−1)D((T k−1)′)[∆T ]dΩ.

(4.66)

The first two terms are zero because the argument is not dependent on ∆T . Now the direc-tional derivative of the overlay residual can be written in a similar manner:

DR0(T k−1)[∆T ] =

∫Ω

:0D(f(x)vo)[∆T ]dΩ−

−∫Ω

:

0D(v′o)[∆T ]k(x, T k−1)(T k−1)′dΩ−

−∫Ω

v′oD(k(x, T k−1))[∆T ](T k−1)′dΩ−

−∫Ω

v′o k(x, T k−1)D((T k−1)′)[∆T ]dΩ.

(4.67)

In order to define the remaining terms in above listed equations, one can use the definitionof the directional derivative (see Equation 4.65). Therefore, applying the chain rule thedirectional derivative of the thermal conductivity becomes:

D(k(x, T k−1))[∆T ] =

[d

dεk(x, T k−1 + ε∆T )

]ε=0

=

=

[dk(x, T k−1 + ε∆T )

d(T k−1 + ε∆T)d(T k−1 + ε∆T )

dε

]ε=0

=

=dk

dT

∣∣∣∣Tk−1

∆T.

(4.68)

As the derivative of the temperature field is linear, its Gateaux derivative is trivial:

D((T k−1)′)[∆T ] =

[d

dε(T k−1 + ε∆T )′

]ε=0

= (∆T )′ (4.69)


Inserting Equations 4.68 and 4.69 into Equations 4.66 and 4.67, the following system of linearPDEs is obtained:

−∫Ω

v′bdk

dT

∣∣∣∣Tk−1

(T k−1)′∆T dΩ−∫Ω

v′b k(x, T k−1)(∆T )′ dΩ = −Rb(T k−1) (4.70)

−∫Ω

v′odk

dT

∣∣∣∣Tk−1

(T k−1)′∆T dΩ−∫Ω

v′o k(x, T k−1)(∆T )′ dΩ = −Ro(T k−1). (4.71)

This, together with the equations for the residual evaluated at the previous iteration stepgiven in Equations 4.63 and 4.64, needs to be solved.

Equations 4.70 and 4.71 are linear in ∆T . The total solution is represented as the sum of baseand overlay temperature solutions (see Equation 3.1). Applying the hp− d idea to Equation4.70 and discretizing it, the Equations 4.70 and 4.71 become:

∫Ω

BTb

dk

dT

∣∣∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NbdΩ ∆Tb

+

∫Ω

BTb

dk

dT

∣∣∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NodΩ ∆To

+

∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bb dΩ ∆Tb

+

∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bo dΩ ∆To =

∫Ω

NTb f(x) dΩ (4.72)

−∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bb dΩ Tk−1

b

−∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bo dΩ Tk−1

o .

Analogously, the discretized equation for the overlay system is obtained:


∫Ω

BTo

dk

dT

∣∣∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NbdΩ ∆Tb

+

∫Ω

BTo

dk

dT

∣∣∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NodΩ ∆To

+

∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bb dΩ ∆Tb

+

∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bo dΩ ∆To =

∫Ω

NTo f(x) dΩ (4.73)

−∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bb dΩ Tk−1

b

−∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bo dΩ Tk−1

o .

Using a more compact notation, Equations 4.72 and 4.73 can be rewritten as follows:

(Kk−1bb + K′k−1

bb

)∆Tb +

(Kk−1bo + K′k−1

bo

)∆To = Fb −Kk−1

bb Tk−1b −Kk−1

bo Tk−1o (4.74)(

Kk−1oo + K′k−1

oo

)∆To +

(Kk−1ob + K′k−1

ob

)∆Tb = Fo −Kk−1

oo Tk−1o −Kk−1

ob Tk−1b . (4.75)

In matrix form, one can obtain:

[Jk−1bb Jk−1

bo

Jk−1ob Jk−1

oo

] [∆Tb

∆To

]=

[Rk−1b

Rk−1o

]. (4.76)

Equation 4.76 represents linear system of equations. Therefore, a linear Gauss-Seidel methodcan be applied to solve it iteratively. For convenience, index l = 1..M will define the counterfor the Gauss-Seidel iterations:

Jk−1bb ∆Tl

b = Rk−1b − Jk−1

bo ∆Tl−1o (4.77)

Jk−1oo ∆Tl

o = Rk−1o − Jk−1

ob ∆Tlb (4.78)

Table 4.8 shows a summary of all the terms occurring in Equations 4.77 and 4.78.

Method 2: Modified Newton-Gauss-Seidel approach

Having a close look to Equations 4.77 and 4.78, it is clear, that the information about ∆Toand ∆Tb can be used as soon as it becomes available after each Gauss-Seidel iteration. Usingthis, the Jacobian matrices can be recomputed in each iteration. Therefore, the equations tobe solved are formulated as follows:


Jk−1,l−1bb ∆Tl

b = Rk−1,l−1b − Jk−1,l−1

bo ∆Tl−1o (4.79)

Jk−1,loo ∆Tl

o = Rk−1,lo − Jk−1,l

ob ∆Tlb, (4.80)

where all the components are as defined in Table 4.8. The system matrices are recomputed,using currently available information about the increment of the overlay or the base solutionrespectively. For example, the base and overlay stiffness matrices are calculated as follows:

Kk−1,l−1bb =

∫Ω

BTb k(x,Nb

(Tk−1b + ∆Tl−1

b

)+ No

(Tk−1o + ∆Tl−1

o

))Bb dΩ

Kk−1,loo =

∫Ω

BTo k(x,Nb

(Tk−1b + ∆Tl

b

)+ No

(Tk−1o + ∆Tl−1

o

))Bo dΩ.

(4.81)


Term Symbol Definition

Base Jacobianmatrix

Jk−1bb

(Kk−1bb + K′k−1

bb

)Coupling Jacobian

matrixJk−1bo

(Kk−1bo + K′k−1

bo

)Overlay Jacobian

matrixJk−1oo

(Kk−1oo + K′k−1

oo

)Coupling Jacobian

matrixJk−1ob

(Kk−1ob + K′k−1

ob

)Base residual vector Rk−1

b Fb −Kk−1bb Tk−1

b −Kk−1bo Tk−1

o

Overlay residualvector

Rk−1o Fo −Kk−1

oo Tk−1o −Kk−1

ob Tk−1b

Base stiffness matrix Kk−1bb

∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bb dΩ

Derived basestiffness matrix

K′k−1bb

∫Ω

BTb

dkdT

∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NbdΩ

Coupling stiffnessmatrix

Kk−1bo

∫Ω

BTb k(x,NbT

k−1b + NoT

k−1o )Bo dΩ

Derived couplingstiffness matrix

K′k−1bo

∫Ω

BTb

dkdT

∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NodΩ

Overlay stiffnessmatrix

Kk−1oo

∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bo dΩ

Derived overlaystiffness matrix

K′k−1oo

∫Ω

BTo

dkdT

∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NodΩ


Kk−1ob

∫Ω

BTo k(x,NbT

k−1b + NoT

k−1o )Bb dΩ


K′k−1ob

∫Ω

BTo

dkdT

∣∣(NbT

k−1b +NoT

k−1o )

(BbTk−1b + BoT

k−1o )NbdΩ

Base source vector Fb

∫Ω

NTb f(x) dΩ

Overlay sourcevector

Fo

∫Ω

NTo f(x) dΩ

Table 4.8: Multiscale hp− d finite element matrices for nonlinear stationary heat equation approxi-mated by Newton-Gauss-Seidel procedure

Method 3: m-step Gauss-Seidel-Newton approach

The m-step Gauss-Seidel-Newton approach can be derived in a similar manner as above.The nonlinear Gauss-Seidel method can be interpreted in terms of solving the i-th nonlinearequation of the system for xi with the other n−1 variables kept fixed [Ortega and Rheinboldt,2000]:

fi(xl1, ..., x

li−1, xi, x

l−1i+1, ..., x

l−1n ) = 0. (4.82)

As the solution of Equation 4.82 is usually not available, an one-dimensional root finding


method is applied [Vrahatis et al., 2003]. In the scope of current work the m-step Newtonmethod is applied.

In order to derive the m-step Gauss-Seidel-Newton equations for the specific nonlinear prob-lem 4.58, first, the hp− d concept is applied (see Equation 3.1):

Rb(To, Tb) = F (vb)−A (To + Tb, k(To + Tb), vb)!

= 0 ∀vb ∈ Vb (4.83)

Ro(To, Tb) = F (vo)−A (To + Tb, k(To + Tb), vp)!

= 0 ∀vo ∈ Vo. (4.84)

The nonlinear Gauss-Seidel approach can also be directly applied at the level of nonlinearPDEs [Langtangen, 1999]. In Equation 4.82, xi represents the variable used in the system ofequations. In the case considered here, the variables are Tb and To and the equations fi areRb and Ro. Applying the nonlinear Gauss-Seidel concept yields:

Rb(Tl−1o , T lb) = F (vb)−A (T l−1

o + T lb , k(T l−1o + T lb), vb) = 0 ∀vb ∈ Vb (4.85)

Ro(Tlo, T

lb) = F (vo)−A (T lo + T lb , k(T lo + T lb), vp) = 0 ∀vo ∈ Vo. (4.86)

where in the first equation the variable T l−1o (x) is kept fixed, while for the overlay equation

just computed T lb(x) is used. As the system of Equations 4.85 and 4.86 is nonlinear andtherefore can not be solved directly, a Newton method is applied to linearize the equations.For the base residual is linearized around the increment of ∆Tb as the overlay temperaturefunction is kept fixed. Omitting high order terms, the following equation is obtained:

Rb(Tl−1o , T l,kb ) = Rb(T

l−1o , T l,k−1

b ) +DRb(Tl−1o , T l,k−1

b )[∆Tb]. (4.87)

Analogously for the overlay residual:

Ro(Tl,ko , T lb) = Ro(T

l,k−1o , T lb) +DRo(T

l,k−1o , T lb)[∆To], (4.88)

where l = 1...M is the counter for the outer Gauss-Seidel iterations and k = 1...N is thecounter for the inner Newton iterations.

As the residuals for both, the overlay and base meshes are required to be zero, Equations4.87 and 4.88 can be written as follows:

DRb(Tl−1o , T l,k−1

b )[∆Tb] = −Rb(T l−1o , T l,k−1

b ) (4.89)

DRo(Tl,k−1o , T lb)[∆To] = −Ro(T l,k−1

o , T lb), (4.90)

where:


Rb(Tl−1o , T l,k−1

b ) =

∫Ω

f(x)vb dΩ−∫Ω

v′bk(x, T l−1o + T l,k−1

b )(T l−1o + T l,k−1

b )′ dΩ (4.91)

Ro(Tl,k−1o , T lb) =

∫Ω

f(x)vo dΩ−∫Ω

v′ok(x, T l,k−1o + T lb)(T

l,k−1o + T lb)

′ dΩ. (4.92)

Using the product rule of the directional derivative yields:

DRb(Tl−1o , T l,k−1

b )[∆Tb] =

∫Ω

:0

D(f(x)vb)[∆Tb] dΩ

−∫Ω

:0D(v′b)[∆Tb]k(x, T l−1

o + T l,k−1b )(T l−1

o + T l,k−1b )′ dΩ

−∫Ω

v′bD(k(x, T l−1o + T l,k−1

b ))[∆Tb](Tl−1o + T l,k−1

b )′ dΩ

−∫Ω

v′bk(x, T l−1o + T l,k−1

b )D(T l−1o + T l,k−1

b )′[∆Tb] dΩ.

(4.93)

Analogously the overlay residual is derived:

DRo(Tl,k−1o , T lb)[∆To] =

∫Ω

:0D(f(x)vo)[∆To] dΩ

−∫Ω

:0D(v′o)[∆To]k(x, T l,k−1

o + T lb)(Tl,k−1o + T lb)

′ dΩ

−∫Ω

v′oD(k(x, T l,k−1o + T lb))[∆To](T

l−1o + T l,k−1

b )′ dΩ

−∫Ω

v′ok(x, T l,k−1o + T lb)D(T l,k−1

o + T lb)′[∆To] dΩ.

(4.94)

Applying the definition of the directional derivative stated in Equation 4.65 and using thechain rule, the terms of Equation 4.93 and 4.94 become:


D(k(x, T l−1o + T l,k−1

b ))[∆Tb] =d

dε

[k(x, T l−1

o + T l,k−1b + ε∆Tb)

]ε=0

=dk

dT∆Tb (4.95)

D(k(x, T l,k−1o + T lb))[∆To] =

d

dε

[k(x, T l,k−1

o + T lb + ε∆To)]ε=0

=dk

dT∆To (4.96)

D(T l−1o + T l,k−1

b )′[∆Tb] =d

dε

[(T l−1o + T l,k−1

b + ∆Tb)′]ε=0

= ∆T ′b (4.97)

D(T l,k−1o + T lb)

′[∆To] =d

dε

[(T l,k−1o + T lb + ∆To)

′]ε=0

= ∆T ′o. (4.98)

Discretization of Equations 4.89 and 4.90 yields:

∫Ω

BTb

dk

dT

∣∣∣∣(NbT

l,k−1b +NoT

l−1o )

(BbTl,k−1b + BoT

l−1o )NbdΩ ∆Tb

+

∫Ω

BTb k(x,NbT

l,k−1b + NoT

l−1o )Bb dΩ ∆Tb =

∫Ω

NTb f(x) dΩ (4.99)

−∫Ω

BTb k(x,NbT

l,k−1b + NoT

l−1o )Bb dΩ Tl,k−1

b

−∫Ω

BTb k(x,NbT

l,k−1b + NoT

l−1o )Bo dΩ Tl−1

o ,

∫Ω

BTo

dk

dT

∣∣∣∣(NbT

lb+NoT

l,k−1o )

(BbTlb + BoT

l,k−1o )NodΩ ∆To

+

∫Ω

BTo k(x,NbT

lb + NoT

l,k−1o )Bo dΩ ∆To =

∫Ω

NTo f(x) dΩ (4.100)

−∫Ω

BTo k(x,NbT

lb + NoT

l,k−1o )Bb dΩ Tl

b

−∫Ω

BTo k(x,NbT

lb + NoT

l,k−1o )Bo dΩ Tl,k−1

o .

In the compact notation one can write:

Jbb(Tl,k−1b ,Tl−1

o )∆Tb = Rb(Tl,k−1b ,Tl−1

o ) (4.101)

Joo(Tlb,T

l,k−1o )∆To = Ro(T

lb,T

l,k−1o ) (4.102)

Table 4.10 shows a summary of formulas for all the terms of Equations 4.101 and 4.102.



Base Jacobian matrix Jbb(Tl,k−1

b ,Tl−1o )

(Kbb(T

l,k−1b ,Tl−1

o ) + K′bb(Tl,k−1b ,Tl−1

o ))

Overlay Jacobianmatrix

Joo(Tl

b,Tl,k−1o )

(Koo(T

lb,T

l,k−1o ) + K′oo(T

lb,T

l,k−1o )

)Base residual Rb Fb −Kbb(T

l,k−1b ,Tl−1

o )Tl,k−1b

vector (Tl,k−1b ,Tl−1

o ) −Kbo(Tl,k−1b ,Tl−1

o )Tl−1o

Overlay residual Ro Fo −Koo(Tlb,T

l,k−1o )Tl,k−1

o

vector (Tlb,T

l,k−1o ) −Kob(T

lb,T

l,k−1o )Tl

b

Base stiffness matrix Kbb

(Tl,k−1b ,Tl−1

o )

∫Ω

BTb k(x,NbT

l,k−1b + NoT

l−1o )Bb dΩ

Derived base K′bb∫Ω

BTb

dkdT

∣∣(NbT

l,k−1b +NoT

l−1o )

(BbTl,k−1b + BoT

l−1o )NbdΩ

stiffness matrix (Tl,k−1b ,Tl−1

o )

Base couplingstiffness matrix

Kbo

(Tl,k−1b ,Tl−1

o )

∫Ω

BTb k(x,NbT

l,k−1b + NoT

l−1o )Bo dΩ


Koo

(Tlb,T

l,k−1o )

∫Ω

BTo k(x,NbT

lb + NoT

l,k−1o )Bo dΩ

Derived overlay K′oo∫Ω

BTo

dkdT

∣∣(NbT

lb+NoT

l,k−1o )

(BbTlb + BoT

l,k−1o )NodΩ

stiffness matrix (Tlb,T

l,k−1o )

Overlay couplingstiffness matrix

Kob

(Tlb,T

l,k−1o )

∫Ω

BTo k(x,NbT

lb + NoT

l,k−1o )Bb dΩ


∫Ω

NTb f(x) dΩ

Overlay source vector Fo

∫Ω

NTo f(x) dΩ

Table 4.10: Multiscale hp − d finite element matrices for nonlinear heat equation approximated byGauss-Seidel-Newton procedure

Method 4: one-step Gauss-Seidel-Newton approach

The formulas for the m-step-Gauss-Seidel method become rather cumbersome, therefore, inpractice often a one-step Gauss-Seidel-Newton method is applied. According to [Ortega andRheinboldt, 2000], the explicit form of one-step Gauss-Seidel-Newton method is as follows:

xli = xl−1i − fi(x

l−1,i)

∂ifi(xl−1,i), (4.103)

where:

xl−1,i = (xl1, ..., xli−1, x

l−1i , xl−1

i+1, ..., xl−1n ). (4.104)

The concept of the one-step approach is to reduce the number of the inner Newton iterationsto one, which will slow down the convergence of the outer Gauss-Seidel iterations. As thederivations of the one-step approach are performed in a similar manner as the m-step Gauss-


Seidel-Newton method, Equation 4.104 will be applied directly to the considered nonlinearheat problem. The system of equations to be solved is as follows:

Jbb(Tl−1b ,Tl−1

o )∆Tb = Rb(Tl−1b ,Tl−1

o ) (4.105)

Joo(Tlb,T

l−1o )∆To = Ro(T

lb,T

l−1o ), (4.106)

where l = 1..N is the mixed Gauss-Seidel-Newton counter.

The terms from Equations 4.105 and 4.106 are very similar to the terms for the m-step Gauss-Seidel-Newton approach (see Table 4.12). The difference is that now the current Newton stepis always taken from the previous Gauss-Seidel iteration.


Base Jacobian matrix Jbb(Tl−1b ,Tl−1

o )(Kbb(T

l−1b ,Tl−1

o ) + K′bb(Tl−1b ,Tl−1

o ))

Overlay Jacobian matrix Joo(Tlb,T

l−1o )

(Koo(T

lb,T

l−1o ) + K′oo(T

lb,T

l−1o )

)Base residual vector Rb(T

l−1b ,Tl−1

o )Fb −Kbb(T

l−1b ,Tl−1

o )Tl−1b

−Kbo(Tl−1b ,Tl−1

o )Tl−1o

Overlay residual vector Ro(Tlb,T

l−1o )

Fo −Koo(Tlb,T

l−1o )Tl−1

o

−Kob(Tlb,T

l−1o )Tl

b

Base stiffness matrix Kbb(Tl−1b ,Tl−1

o )∫Ω

BTb k(x,NbT

l−1b + NoT

l−1o )Bb dΩ

Derived baseK′bb(T

l−1b ,Tl−1

o )

∫Ω

BTb

dkdT

∣∣(NbT

l−1b +NoT

l−1o )

stiffness matrix (BbTl−1b + BoT

l−1o )NbdΩ

Coupling stiffness matrix Kbo(Tl−1b ,Tl−1

o )∫Ω

BTb k(x,NbT

l−1b + NoT

l−1o )Bo dΩ

Overlay stiffness matrix Koo(Tlb,T

l−1o )

∫Ω

BTo k(x,NbT

lb + NoT

l−1o )Bo dΩ

Derived overlayK′oo(T

lb,T

l−1o )

∫Ω

BTo

dkdT

∣∣(NbT

lb+NoT

l−1o )

stiffness matrix (BbTlb + BoT

l−1o )NodΩ

Coupling stiffness matrix Kob(Tlb,T

l−1o )

∫Ω

BTo k(x,NbT

lb + NoT

l−1o )Bb dΩ


∫Ω

NTb f(x) dΩ

Overlay source vector Fo

∫Ω

NTo f(x) dΩ

Table 4.12: Multiscale hp− s finite element matrices for a nonlinear heat equation approximated bya one-step Gauss-Seidel-Newton procedure

Both methods can be easily adjusted to m-step Jacobi-Newton and one-step Jacobi-Newtonmethods. The difference lies in the equation for the overlay mesh, where for Jacobi-likemethods the information of new calculated Tb is not used (instead, the result from theprevious Jacobi iteration).


4.3.2 Implementation of the hp− d method

The implementation of Method 1, 2, 3 and 4 is summarized in Algorithm 3, 4, 5 and 6.

Algorithm 3 hp-d algorithm for nonlinear stationary heat problem with linear elements(Method 1)

1: Create base mesh2: Create overlay mesh3: for k(iNewton) = 1 ..maximumNumberOfIterations do4: Set ∆T l−1

b and T l−1o to zero

5: for l(iGaussSeidel) = 1 ..maximumNumberOfIterations do6: Compute base solution increment:7: Compute Kk−1

bb , K′k−1bb , Fb

8: Compute Kk−1bo Tk−1

o ,Kk−1bo ∆Tl−1

o ,K′k−1bo ∆Tl−1

o

9: Solve Equation 4.7710: Compute overlay solution increment:11: Compute Kk−1

oo , K′k−1oo , Fo

12: Compute Kk−1ob Tk−1

b ,Kk−1ob ∆Tl

b,K′k−1bo ∆Tl

b

13: Solve Equation 4.7814: Check convergence of l2-norm of complete solution15: if l2-norm is converged16: break Gauss-Seidel Iterations17: end if18: end for19: Check convergence of Newton iterations20: if converged21: Set Tk

b = Tk−1b + ∆Tl

b, Tko = Tk−1

o + ∆Tlo

22: break Newton Iterations23: end if24: end for25: Postprocess the solution



1: Create base mesh2: Create overlay mesh3: for k(iNewton) = 1 ..maximumNumberOfIterations do4: Set ∆T l−1

b and T l−1o to zero

5: for l(iGaussSeidel) = 1 ..maximumNumberOfIterations do6: Compute base solution increment:7: Compute Kk−1,l−1

bb , K′k−1,l−1bb , Fb

8: Compute Kk−1,l−1bo Tk−1,l−1

o ,Kk−1,l−1bo ∆Tl−1

o ,K′k−1,l−1bo ∆Tl−1

o

9: Solve Equation 4.7910: Update Tk−1,l

b = Tk−1,l−1b + ∆Tl

b

11: Compute overlay solution increment:12: Compute Kk−1,l

oo , K′k−1,loo , Fo

13: Compute Kk−1,lob Tk−1,l

b ,Kk−1,lob ∆Tl

b,K′k−1,lbo ∆Tl

b

14: Solve Equation 4.8015: Update Tk−1,l

o = Tk−1,l−1o + ∆Tl

o

16: Check convergence of l2-norm of complete solution17: if l2-norm is converged18: break Gauss-Seidel Iterations19: end if20: end for21: Check convergence of Newton iterations22: if converged23: Set Tk,l

b = Tk−1,lb , Tk,l

o = Tk−1,lo

24: break Newton Iterations25: end if26: end for27: Postprocess the solution



1: Create base mesh2: Create overlay mesh3: for l(iGaussSeidel) = 1 ..maximumNumberOfIterations do4: for k(iNewton) = 1 ..maximumNumberOfIterations do5: Compute base solution increment:6: Compute Kbb(T

l,k−1b ,Tl−1

o ), K′bb(Tl,k−1b ,Tl−1

o ), Fb

7: Compute Kbo(Tl,k−1b ,Tl−1

o )Tl−1o

8: Solve Equation 4.1019: Update Tl,k

b = Tl,k−1b + ∆Tk

b

10: Check convergence of Newton iterations11: if converged12: Set Tl

b = Tl,kb

13: break Newton Iterations14: end for15: for k(iNewton) = 1 ..maximumNumberOfIterations do16: Compute overlay solution increment:17: Compute Koo(T

lb,T

l,k−1o ), K′oo(T

lb,T

l,k−1o ), Fo

18: Compute Kob(Tlb,T

l,k−1o )Tl,k−1

b

19: Solve Equation 4.10220: Update Tl,k

o = Tl,k−1o + ∆Tk

o

21: Check convergence of Newton iterations22: if converged23: Set Tl

o = Tl,ko

24: break Newton Iterations25: end for26: Check convergence of l2-norm of complete solution27: if l2-norm is converged28: break Gauss-Seidel Iterations29: end if30: end for31: Postprocess the solution



1: Create base mesh2: Create overlay mesh3: for l(iGaussSeidelNewton) = 1 ..maximumNumberOfIterations do4: Compute base solution increment:5: Compute Kbb(T

l−1b ,Tl−1

o ), K′bb(Tl−1b ,Tl−1

o ), Fb

6: Compute Kbo(Tl−1b ,Tl−1

o )Tl−1o

7: Solve Equation 4.1058: Update Tl

b = Tl−1b + ∆Tl

b

9: Compute overlay solution increment:10: Compute Koo(T

lb,T

l−1o ), K′oo(T

lb,T

l−1o ), Fo

11: Compute Kob(Tlb,T

l−1o )Tl

b

12: Solve Equation 4.10613: Update Tl

o = Tl−1o + ∆Tl

o

14: Check convergence of l2-norm of complete solution15: if l2-norm is converged16: break Gauss-Seidel-Newton Iterations17: end if18: end for19: Postprocess the solution

It is important to mention, that all matrices within the overlay zone are computed with thehelp of composed Gauss quadrature rule. The L2-error of the complete hp−d solution is usedas the breaking criteria for Gauss-Seidel iterations. The convergence of the Newton scheme isassessed by computing the inner product between residual and increment of the solution. Asthe complete residual is not available at any point of computation, this condition is replacedby two simultaneously fulfilled criteria for base and overlay calculations. For the residual thewhole constrained right-hand side of Equations 4.77 and 4.78 for Method 1, Equations 4.79and 4.80 for Method 2, Equations 4.101 and 4.102 for Method 3 and Equations 4.105 and4.106 for Method 4 is used. It is necessary to mention, that Method 4 can be easily convertedto a one-step Jacobi-Newton method, by moving the update of the base solution after thecalculation of the local solution increment.

4.3.3 Verification

Comparison with the multilevel hp− solution

As it was done in the sections above, first, the hp − d result is compared to the result ofthe monolithic hp − d code. For verification of the nonlinear problem, three test cases wereconsidered, increasing the complexity.


Test case 1 has the following thermal conductivity:

k(x, T ) = 5T + 1 (4.107)

and the heat source as follows:

f(x) = −4320x2 + 4320− 696. (4.108)

The analytical solution is therefore:

T (x) = −12x2 + 12x (4.109)

With the same analytical solution as in Equation 4.109 the following material parameters areconsidered (Test case 2):

k(x, T ) = sin(T ) + 10 (4.110)

with the heat source as follows:

f(x) = 24 · sin(−12x2 + 12x) + (−576x2 + 576x− 144) · cos(−12x2 + 12x) + 240 (4.111)

In test case 3 the Gaussian bell is tested again (see Section 4.2). The analytical solution is:

T (x) = exp

(−(x− 0.2)2

2 · 0.022

). (4.112)

For Test case 3 the thermal diffusivity is chosen as:

k(x, T ) = 4T + 4x+ 40 (4.113)

For all three cases a base mesh of 40 elements is considered. The overlay mesh consists oftwo elements per overlayed base elements. Moreover, the number of iterations required toachieve convergence is stored. Table 4.13 shows the comparison of the results consideringfull overlay, Table 4.14 summarizes the results for partial overlay. It is observed, that allresults are providing only numerical difference in comparison with the solution obtainedby monolithical multilevel hp code. The tolerance when the iterations are stopped for allmethods are set as 10−10.


Case Method 1 Method 2 Method 3 Method 4

1

e = 1.2232 · 10−28 e = 3.8333 · 10−29 e = 3.5879 · 10−28 e = 3.3809 · 10−29

8 Newtoniterations

1 Newtoniteration

2 GS iterations9 mergediterations

1-8: 2 GS 9 GS1: 8 and 3

Newton; 2: 1 and1

2

e = 5.4774 · 10−30 e = 5.2854 · 10−29 e = 6.8014 · 10−30 e = 2.6607 · 10−29

4 Newtoniterations

1 Newtoniteration

2 GS iterations5 mergediterations1: 2 GS; 2: 3 GS;

3-4: 2 GS5 GS

1: 4 and 3Newton; 2: 1 and

1

3

e = 1.0175 · 10−26 e = 1.2581 · 10−26 e = 1.1031 · 10−26 e = 9.7419 · 10−27

4 Newtoniterations

1 Newtoniteration

4 GS iterations8 mergediterations1: 3 GS; 2-5: 4

GS6 GS


2; 3-4: 1 and 1

Table 4.13: Verification of the multiscale hp − d code with the multilevel hp−code for nonlinearstationary test cases considering a full overlay


Case Method 1 Method 2 Method 3 Method 4

1

Partial overlay: Ωo = [0.3, 0.6]

e = 3.9916 · 10−28 e = 2.3095 · 10−28 e = 2.5028 · 10−29 e = 7.9233 · 10−29

8 Newtoniterations

1 Newtoniteration


1-8: 2 GS 9 GS1: 8 and 3

Newton; 2: 1 and1

2


e = 6.6404 · 10−29 e = 1.9630 · 10−30 e = 1.5258 · 10−29 e = 2.9844 · 10−30

4 Newtoniterations

1 Newtoniteration


1-4: 2 GS 5 GS1: 4 and 3

Newton; 2: 1 and1

3


e = 9.1641 · 10−27 e = 1.4827 · 10−26 e = 8.2118 · 10−27 e = 9.8324 · 10−27

4 Newtoniterations

1 Newtoniteration

4 GS iterations5 mergediterations1: 3 GS; 2-4: 4

GS6 GS


2; 3-4: 1 and 1

Table 4.14: Verification of the multiscale hp − d code with the multilevel hp-code for nonlinearstationary test cases considering a partial overlay

Convergence studies

The convergence studies are performed by constantly increasing the number of overlayedelements per base elements. The studies were started considering ten base elements. Allconvergence studies are performed only on the full overlay, as Test Case 1 and 2 wouldrequire the control on two parameters at the same time: the number of base and overlayedelements. For all four methods the number of overlay elements per base element is constantlyincreased from two to hundred. The relative L2-error of the numerical solution in comparisonwith the analytical one (See Equation 4.109) is used as the convergence criteria. The toleranceis set to 10−10. Figure 4.22 shows the convergence graph for test case 1 and Figure 4.23 fortest case 2. As it can be seen, that the rate of convergence of all four methods in space isidentical. The difference between the obtained L2-errors is only of order of 10−19. The resultsof the convergence studies are summarized in Table C.1 and Table C.2.


10 20 30 40 50 60 70 80 90 100

10−12

10−11

10−10

10−9

10−8

10−7

10−6


L2

erro

r,[

-]

Full overlay method 1Full overlay method 2Full overlay method 3Full overlay method 4

Figure 4.22: Convergence studies on the spatial refinement for Test case 1

10 20 30 40 50 60 70 80 90 100

10−12

10−11

10−10

10−9

10−8

10−7

10−6


L2

erro

r,[

-]



The convergence studies for test case 3 was performed on the partial overlay within [0.1; 0.3].


The number of base elements is considered as 40 in order to capture the physical effect.At first, the number of the overlayed elements per base element was increased from two tohundred. Figure 4.24 shows that the rate of convergence of all the methods is the same, butthe result is converging to the error about 10−8. When the error levels off, the error in thenumerical solution is originating from the rest of the non-overlayed domain. Therefore, forfurther studies the number of overlayed elements per base element is fixed as 50, while thenumber of base elements is increased from 50 to 100. Figure 4.25 shows that the numericalsolution is converging to the analytical one. The results can be found in Table C.3 and C.4.

10 20 30 40 50 60 70 80 90 100

10−8

10−7

10−6

10−5

10−4


L2

erro

r,[

-]




40 45 50 55 60 65 70 75 80 85 90 95 10010−11

10−10

10−9

10−8

Number of base elements

L2

erro

r,[

-]


Figure 4.25: Convergence studies on the spatial refinement for Test case 3, keeping the number ofthe overlayed elements as 50

It is concluded, that all four methods have the same convergence rate while refining in space.Therefore, the questions of the cost arises. One measure would be to count the number ofiterations necessary to reach a certain tolerance in the L2-error.

Cost studies

As the test cases are one-dimensional, the problems are rather small and therefore the runtimeof the program cannot be used as a cost estimate. In realistic 3D problems with a highnumber of degrees of freedom, the solution of the linear systems is likely to be the bottleneck.Therefore, a good way to extrapolate the cost from 1D problems is to count the number oftimes a linear system is solved until the algorithm has reached a certain tolerance in the L2

error of the solution. The drawback is that the sizes of the arising systems from base andoverlay problems is different. As the cost function for solving a linear system of equationshighly depends on the matrix type and the implementation the respective weights can not bedetermined in a general way. However, as in most settings the number of base and overlaysolutions are similar, this approach is considered to be an appropriate measure in the contextof this thesis.

Test case 1 is considered with the full overlay over ten base elements. For this case, eachbase element is overlayed with thirty elements. For Method 1 the convergence criteria forthe outer (Newton) iteration is the inner product between base residual and the increment ofthe base solution together with the inner product of the overlay solution and the incrementin the overlay solution. As in this case only these parameters are possible to compare, therespective inner product is plotted in Figure 4.26. As for method 2, only one Newton iterationis needed, the relative error in the L2-norm of the numerical solution in comparison to the


previously calculated one is recorded after each Gauss-Seidel iteration within the Newtonloop. In Method 3 the relative error in the L2-norm is recorded after each outer Gauss-Seideliteration. As in the Method 4 only one iteration loop is implemented, the error is recordedafter each merged iteration.

100 100.2 100.4 100.6 100.8 101 101.2 101.410−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Number of calls to ”\”

Rel

ativ

eL

2er

ror,

[-

]

Method 1Method 2Method 3Method 4

Figure 4.26: Number of calls to “\” against the convergence criteria for Test case 1 with full overlay

It is clear, that Method 3 and Method 4 converge faster than Method 1 and Method 2.Moreover, it is shown, that Method 3 has a faster rate of convergence compared to Method 4.Now, it is interesting to note, how this conclusion changes with the considered Test case. Testcase 2 is studied with the same configuration as the test one: ten base elements, overlayedwith 30 elements each. Figure 4.27 shows that Method 3 gives again a faster convergencethan Method 4. Originally, the idea of the m-step method (Method 3) is that it converges m-times faster than the one-step method (Method 4), where m is the number of the consideredNewton steps. As in the scope of the current work, m is not fixed, but is different in eachiteration, it is hard to evaluate how much faster this method should theoretically converge forthe considered case. Method 2 is better than Method 1, while still the third and the fourthmethod perform best.


100 100.2 100.4 100.6 100.8 101 101.210−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


Rel

ativ

eL

2er

ror,

[-

]



100.7 100.8 100.9 101 101.1 101.2 101.3 101.4 101.510−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1


Rel

ati

veL

2er

ror,

[-

]




In order to keep the conditions the same, now the same test is performed with Test case 3with a full overlay of ten base elements with 30 overlay elements each. Figure 4.28 showsthat now the fourth method converges faster than Method 3.

The application of a partial overlay had no influence on the rate of convergence of the methods.Therefore, it can be seen that Methods 4 and 3 are performing better for all of the test casescompared to the first and the second method. Considering the implementational costs, theydo not require the implementation of coupling terms involving the increment of the solution.Thus, these two methods are considered to be the most efficient from the implementationaland convergence point of view. It is interesting to note, that from the implementational pointof view, Method 1 has also advantages, as all of the system matrices are dependent only onthe previous Newton step, they can be cached while performing the Gauss-Seidel loop. Forthe rest of the methods the matrices have to be recomputed in each iteration, which does notprovide the opportunity for saving computational costs.

4.3.4 Significance of the precision of the Jacobian computation and thecoupling terms

As one is aiming for the best balance between implementational costs and convergence char-acteristics of the solution, it is decided to check, how the precision of the computation of theJacobian matrices is influencing these characteristics. As it can be seen in Tables 4.8, 4.10and 4.12, the Jacobian matrices for the base and overlay meshes are consisting of two terms:one, originating from the thermal conductivity Kbb and Koo, and the other one from thederivative of the thermal conductivity K′bb and K′oo. When any of the terms is neglected, theconvergence of the numerical solution should still be reached. For Methods 1 and 2 the terms,originating from the derivative of the thermal diffusivity, are also present in the right-handside and, therefore, should be eliminated for these studies.

First, all studies are performed, using Method 1. Test case 1 is considered with ten baseelements. The convergence tolerance for both, Newton and Gauss-Seidel iterations, is 10−10.The number of overlay elements per base element is constantly increased from two to hundred.Table C.5 and C.6 show, that the precision of the computation of the Jacobian has no influenceon the quality of the solution. However it is visible, that the number of necessary iterationsis increasing.

Method 2 is tested in the same manner. As it is shown in Table C.7 and Table C.8, the relativeL2-error of the numerical solution with the analytical solution is slightly different comparedthe Method 1. But it can be observed that more iterations are necessary to reach convergence.It is clear, that not precise prediction of the Jacobian leads to a slower convergence. Thesame is observed for all test cases.

The significance of the extra terms is evaluated in the same manner as before. The followingdefinitions are used:


SoK =‖Kbo (To) ‖L2

‖Fb‖L2

(4.114)

SbK =‖Kob (Tb) ‖L2

‖Fo‖L2

(4.115)

SoK∆T =‖Kbo (∆To) ‖L2

‖Fb‖L2

(4.116)

SbK∆T =‖Kob (∆Tb) ‖L2

‖Fo‖L2

(4.117)

SoK′∆T =‖K′bo (∆To) ‖L2

‖Fb‖L2

(4.118)

SbK′∆T =‖K′ob (∆Tb) ‖L2

‖Fo‖L2

. (4.119)

Method 1 was tested on the significance of extra coupling terms, coming to the right handside. In table C.13 it is shown, that some of the terms can be neglected. For this case allthe coupling terms except for KboTo and KobTb are set to zero and a convergence analysisis performed. In table C.14 it can be seen, that the same quality of the solution comparedto the case with all coupling terms is reached.

Table C.15 shows the significance of the extra term with respect to the performed Gauss-Seidel iteration within the Newton iteration in Method 2 for the Test Case 1 with ten baseelements and 30 overlay elements each. It is clear, that the terms KboTo and KobTb cannotbe neglected. The rest of the terms are set to zero and the same convergence analysisis performed. Table C.16 summarizes the result of the convergence test. The numericalsolution is converging to the numerical one and almost the same number of iterations isneeded. Therefore, extra coupling terms, originating form the increment of the solution, canbe neglected. For all test cases the result appears to be the same.

For Method 3 and 4, the same studies for neglecting the prime terms in the computation ofthe Jacobian are performed. The same conclusion as above can be made, as the number ofnecessary iterations is increased. Table C.9 shows the convergence results for Method 3 withthe complete formulation and Table C.10, neglecting the prime terms in the Jacobian. TablesC.11 and C.12 summarize the results for Method 4.

Regarding the extra terms, now only two are relevant (see Equation 4.114 and 4.184). Theresults of Method 3 are shown in Table C.17 and for Method 4 in Table C.18. It is obvious,that both coupling terms are important for the solution. Therefore, tests without them arenot performed. Extra coupling terms for these two cases cannot be neglected.

4.4 Nonlinear transient heat equation

For the nonlinear transient case, all terms in Equation 2.7 are considered. First, numericalapproaches to decouple and solve the governing equation are introduced. Then, specialimplementational aspects are pointed out. The verification is performed using the resultsfrom multilevel hp− calculations.

4.4. Nonlinear transient heat equation 77


In this section, the overlay approximation is applied to the one-dimensional nonlinear tran-sient heat equation. Similar to Section 4.3 three main methods will be derived: Newton-Gauss-Seidel (Method 1), m-step Gauss-Seidel-Newton (Method 2) and one-step Gauss-Seidel-Newton (Method 3).

Method 1: Newton-Gauss-Seidel approach

At first, the Backward-Euler discretization (see Section 2.4) will be applied on the PDE level[Langtangen, 2015]. The approximation of the time derivative of the temperature function istherefore:

Tn − Tn−1

∆t= f(x, Tn), (4.120)

where index n indicates the time step. Equation 4.120 can be inserted in Equation 2.1, whichyields:

ρ(x, Tn)c(x, Tn)Tn − Tn−1

∆t− ∂

∂x

(k(x, Tn)

∂Tn∂x

)= f(x, t). (4.121)

As Equation 4.121 is solved for ∆Tn = Tn − Tn−1, it can be reorganized:

ρ(x, Tn)c(x, Tn)∆Tn∆t− ∂

∂x

(k(x, Tn)

∂Tn−1

∂x

)− ∂

∂x

(k(x, Tn)

∂(∆Tn)

∂x

)= f(x, t). (4.122)

Equation 4.122 is the nonlinear stationary PDE, which should be solved for each time stepn. In order to solve it, a linearization can be applied. Newton’s method is normally usedto linearize the system of algebraic equations. It can be applied also at the PDE level.Assuming that ∆T k−1

n is the approximation of the unknown ∆Tn, then by Newton methoda better approximation can be obtained:

∆Tn = ∆T k−1n + δT (4.123)

where k is indicating the index for Newton iterations. Inserting Equation 4.123 into Equation4.122 yields:


ρ(x, Tn−1 + ∆T k−1n + δT )c(x, Tn−1 + ∆T k−1

n + δT )∆T k−1

n + δT

∆t

− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n + δT )∂Tn−1

∂x

)− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n + δT )∂(∆T k−1

n + δT )

∂x

)= f(x, t).

(4.124)

Then the update of the temperature field for the next Newton iteration is obtained by Equa-tion 4.125.

∆T kn = ∆T k−1n + δT. (4.125)

The nonlinearities in the material parameters can be expanded using Taylor expansion (see[Langtangen, 2015]). Up to the first order:

k(x, Tn−1 + ∆T k−1

n + δT)

= k(x, Tn−1 + ∆T k−1

n

)+dk

dT

(x, Tn−1 + ∆T k−1

n

)δT +O

(δT 2

)≈ k

(x, Tn−1 + ∆T k−1

n

)+dk

dT

(x, Tn−1 + ∆T k−1

n

)δT

(4.126)

ρ(x, Tn−1 + ∆T k−1

n + δT)

= ρ(x, Tn−1 + ∆T k−1

n

)+dρ

dT

(x, Tn−1 + ∆T k−1

n

)δT +O

(δT 2

)≈ ρ

(x, Tn−1 + ∆T k−1

n

)+dρ

dT

(x, Tn−1 + ∆T k−1

n

)δT

(4.127)

c(x, Tn−1 + ∆T k−1

n + δT)

= c(x, Tn−1 + ∆T k−1

n

)+dc

dT

(x, Tn−1 + ∆T k−1

n

)δT +O

(δT 2

)≈ c

(x, Tn−1 + ∆T k−1

n

)+dc

dT

(x, Tn−1 + ∆T k−1

n

)δT.

(4.128)

Inserting the linear approximation of the material parameters in Equation 4.124 results in:


(ρ(x, Tn−1 + ∆T k−1

n )c(x, Tn−1 + ∆T k−1n )

) ∆T k−1n + δT

∆t

+

(ρ(x, Tn−1 + ∆T k−1

n )dc

dT

(x, Tn−1 + ∆T k−1

n

)δT

)∆T k−1

n + δT

∆t

+

(c(x, Tn−1 + ∆T k−1

n )dρ

dT

(x, Tn−1 + ∆T k−1

n

)δT

)∆T k−1

n + δT

∆t

− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂x+dk

dT(x, Tn−1 + ∆T k−1

n )δT∂Tn−1

∂x

)− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n + δT )

∂x+dk

dT(x, Tn−1 + ∆T k−1

n )δT∂(∆T k−1

n + δT )

∂x

)=

= f(x, t)

(4.129)

The reorganization of Equation 4.129 leads to the shorter form:

δF(δT,∆T kn

)= −F

(∆T kn

)(4.130)

where

F(

∆T kn

)= ρ(x, Tn−1 + ∆T k−1

n )c(x, Tn−1 + ∆T k−1n )

∆T k−1n

∆t

− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂x

)− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂x

)− f(x, t)

(4.131)

δF(δT,∆T kn

)= ρ(x, Tn−1 + ∆T k−1

n )c(x, Tn−1 + ∆T k−1n )

δT

∆t

+ ρ(x, Tn−1 + ∆T k−1n )

dc

dT(x, Tn−1 + ∆T k−1

n )δT∆T k−1

n

∆t

+ c(x, Tn−1 + ∆T k−1n )

dρ

dT(x, Tn−1 + ∆T k−1

n )δT∆T k−1

n

∆t

− ∂

∂x

(dk

dT(x, Tn−1 + ∆T k−1

n )δT∂Tn−1

∂x

)− ∂

∂x

(k(x, Tn−1 + ∆T k−1

n )∂(δT )

∂x

)− ∂

∂x

(dk

dT(x, Tn−1 + ∆T k−1

n )δT∂(∆T k−1

n )

∂x

)

(4.132)

The hierarchical overlay is constructed such that the linear dependencies between base and


overlay spaces are eliminated (see Equation 4.48). Therefore, using variational calculus andapplying the integration by parts, Equation 4.130 can be rewritten for both base and overlayspaces separately:

δFb

(δT,∆T kn

)= −Fb

(∆T kn

)(4.133)

δFo

(δT,∆T kn

)= −Fo

(∆T kn

)(4.134)

where

Fb

(∆T kn

)=

∫Ω

ρ(x, Tn−1 + ∆T k−1n )c(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tvbdΩ

+

∫Ω

∂vb∂x

k(x, Tn−1 + ∆T k−1n )

∂Tn−1

∂xdΩ

+

∫Ω

∂vb∂x

k(x, Tn−1 + ∆T k−1n )

∂(∆T k−1n )

∂xdΩ−

∫Ω

f(x, t)vbdΩ

−[vbk(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂x

]Γ

−[vbk(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂x

]Γ

(4.135)


δFb

(δT,∆T kn

)=

∫Ω

ρ(x, Tn−1 + ∆T k−1n )c(x, Tn−1 + ∆T k−1

n )δT

∆tvbdΩ

+

∫Ω

ρ(x, Tn−1 + ∆T k−1n )

dc

dT(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tδTvbdΩ

+

∫Ω

c(x, Tn−1 + ∆T k−1n )

dρ

dT(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tδTvbdΩ

+

∫Ω

∂vb∂x

dk

dT(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂xδTdΩ

+

∫Ω

∂vb∂x

k(x, Tn−1 + ∆T k−1n )

∂(δT )

∂xdΩ

+

∫Ω

∂vb∂x

dk

dT(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂xδTdΩ

−[dk

dT(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂xδTvb

]Γ

−[k(x, Tn−1 + ∆T k−1

n )∂(δT )

∂xvb

]Γ

−[dk

dT(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂xδTvb

]Γ

(4.136)

Fo

(∆T kn

)=

∫Ω

ρ(x, Tn−1 + ∆T k−1n )c(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tvodΩ

+

∫Ω

∂vo∂x

k(x, Tn−1 + ∆T k−1n )

∂Tn−1

∂xdΩ

+

∫Ω

∂vo∂x

k(x, Tn−1 + ∆T k−1n )

∂(∆T k−1n )

∂xdΩ−

∫Ω

f(x, t)vodΩ

−[vok(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂x

]Γ

−[vok(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂x

]Γ

(4.137)


δFo

(δT,∆T kn

)=

∫Ω

ρ(x, Tn−1 + ∆T k−1n )c(x, Tn−1 + ∆T k−1

n )δT

∆tvodΩ

+

∫Ω

ρ(x, Tn−1 + ∆T k−1n )

dc

dT(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tδTvodΩ

+

∫Ω

c(x, Tn−1 + ∆T k−1n )

dρ

dT(x, Tn−1 + ∆T k−1

n )∆T k−1

n

∆tδTvodΩ

+

∫Ω

∂vo∂x

dk

dT(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂xδTdΩ

+

∫Ω

∂vo∂x

k(x, Tn−1 + ∆T k−1n )

∂(δT )

∂xdΩ

+

∫Ω

∂vo∂x

dk

dT(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂xδTdΩ

−[dk

dT(x, Tn−1 + ∆T k−1

n )∂Tn−1

∂xδTvo

]Γ

−[k(x, Tn−1 + ∆T k−1

n )∂(δT )

∂xvo

]Γ

−[dk

dT(x, Tn−1 + ∆T k−1

n )∂(∆T k−1

n )

∂xδTvo

]Γ

(4.138)

After discretization of Equations 4.133 and 4.134 and applying the hp− d idea (see Equation3.1), the result can be written in compact notation:

(1

∆tb

(Mk−1

bb,n + M′k−1bb,n + M′k−1

bb,n

)+(Kk−1bb,n + K′k−1

bb,n

))δTb

+

(1

∆to

(Mk−1

bo,n + M′k−1bo,n + M′k−1

bo,n

)+(Kk−1bo,n + K′k−1

bo,n

))δTo =

(4.139)

= Fb −Kk−1bb,n

(Tb,n−1 + ∆Tk−1

b,n

)−Kk−1

bo,n

(To,n−1 + ∆Tk−1

o,n

)−

− 1

∆tbMk−1

bb,n∆Tk−1b,n −

1

∆toMk−1

bo,n∆Tk−1o,n

(1

∆tb

(Mk−1

ob,n + M′k−1ob,n + M′k−1

ob,n

)+(Kk−1ob,n + K′k−1

ob,n

))δTb

+

(1

∆to

(Mk−1

oo,n + M′k−1oo,n + M′k−1

oo,n

)+(Kk−1oo,n + K′k−1

oo,n

))δTo =

(4.140)


= Fo −Kk−1ob,n

(Tb,n−1 + ∆Tk−1

b,n

)−Kk−1

oo,n

(To,n−1 + ∆Tk−1

o,n

)−

− 1

∆tbMk−1

ob,n∆Tk−1b,n −

1

∆toMk−1

oo,n∆Tk−1o,n

As it is clear, the coefficients of time step may differ on the base and overlay mesh, as thesubstepping technique might be applied further. Equations 4.139 and 4.140 can be rewrittenin matrix form:

[Jk−1bb,n Jk−1

bo,n

Jk−1ob,n Jk−1

oo,n

] [δTb

δTo

]=

[Fk−1b,n

Fk−1o,n

](4.141)

Gauss-Seidel method can be applied to the linearized system of equations 4.141. The counterfor Gauss-Seidel iterations l will be defined as before:

Jk−1bb,nδT

lb = Fk−1

b,n − Jk−1bo,nδT

l−1o (4.142)

Jk−1oo,nδT

lo = Fk−1

o,n − Jk−1ob δTl

b (4.143)

All components needed for Equations 4.142 and 4.143 are stated in detail in Table 4.15.



Base Jacobianmatrix

Jk−1bb,n

1∆tb

(Mk−1

bb,n + M′k−1bb,n + M′k−1

bb,n

)+(Kk−1bb,n + K′k−1

bb,n

)Coupling Jacobian

matrixJk−1bo,n

1∆to

(Mk−1

bo,n + M′k−1bo,n + M′k−1

bo,n

)+(Kk−1bo,n + K′k−1

bo,n

)Overlay Jacobian

matrixJk−1oo,n

(1

∆to

(Mk−1

oo,n + M′k−1oo,n + M′k−1

oo,n

)+(Kk−1oo,n + K′k−1

oo,n

))Coupling Jacobian

matrixJk−1ob,n

(1

∆tb

(Mk−1

ob,n + M′k−1ob,n + M′k−1

ob,n

)+(Kk−1ob,n + K′k−1

ob,n

))Base residual vector Fk−1

b,n Fb −Kk−1bb,n

(Tb,n−1 + ∆Tk−1

b,n

)−

Kk−1bo,n

(To,n−1 + ∆Tk−1

o,n

)− 1

∆tbMk−1

bb,n∆Tk−1b,n −

1∆to

Mk−1bo,n∆Tk−1

o,n


Fk−1o,n Fo −Kk−1

ob,n

(Tb,n−1 + ∆Tk−1

b,n

)−

Kk−1oo,n

(To,n−1 + ∆Tk−1

o,n

)− 1

∆tbMk−1

ob,n∆Tk−1b,n −

1∆to

Mk−1oo,n∆Tk−1

o,n

Base stiffness matrix Kk−1bb,n

∫Ω

BTb k(x,NbT

k−1b,n + NoT

k−1o,n )Bb dΩ


K′k−1bb,n

∫Ω

BTb

dkdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(BbTk−1b,n + BoT

k−1o,n )NbdΩ


Kk−1bo,n

∫Ω

BTb k(x,NbT

k−1b,n + NoT

k−1o,n )Bo dΩ


K′k−1bo,n

∫Ω

BTb

dkdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(BbTk−1b,n + BoT

k−1o,n )NodΩ


Kk−1oo,n

∫Ω

BTo k(x,NbT

k−1b,n + NoT

k−1o,n )Bo dΩ


K′k−1oo,n

∫Ω

BTo

dkdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(BbTk−1b,n + BoT

k−1o,n )NodΩ


Kk−1ob,n

∫Ω

BTo k(x,NbT

k−1b,n + NoT

k−1o,n )Bb dΩ


K′k−1ob,n

∫Ω

BTo

dkdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(BbTk−1b,n + BoT

k−1o,n )NbdΩ

Base mass matrix Mk−1bb,n

∫Ω

NTb ρ(x,NbT

k−1b,n + NoT

k−1o,n ) c(x,NbT

k−1b,n +

NoTk−1o,n )Nb dΩ

Derived base massmatrix

M′k−1bb,n

∫Ω

NTb

dρdT

∣∣∣(NbT

k−1b,n +NoT

k−1o,n )

c(x,NbTk−1b,n +

NoTk−1o,n )(Nb∆Tk−1

b,n + No∆Tk−1o,n )NbdΩ


M′k−1bb,n

∫Ω

NTb

dcdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

ρ(x,NbTk−1b,n +

NoTk−1o,n )(Nb∆Tk−1

b,n + No∆Tk−1o,n )NbdΩ

Coupling massmatrix

Mk−1bo,n

∫Ω

NTb ρ(x,NbT

k−1b,n + NoT

k−1o,n )c(x,NbT

k−1b,n +

NoTk−1o,n )No dΩ

Derived couplingmass matrix

M′k−1bo,n

∫Ω

NTb

dρdT

∣∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +

No∆Tk−1o,n )c(x,NbT

k−1b,n + NoT

k−1o,n )NodΩ



M′k−1bo,n

∫Ω

NTb

dcdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +

No∆Tk−1o,n )ρ(x,NbT

k−1b,n + NoT

k−1o,n )NodΩ

Overlay mass matrix Mk−1oo,n

∫Ω

NTo ρ(x,NbT

k−1b,n + NoT

k−1o,n )c(x,NbT

k−1b,n +

NoTk−1o,n )No dΩ

Derived overlay massmatrix

M′k−1oo,n

∫Ω

NTo

dρdT

∣∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +


k−1b,n + NoT

k−1o,n )NodΩ


M′k−1oo,n

∫Ω

NTo

dcdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +


k−1b,n + NoT

k−1o,n )NodΩ

Coupling massmatrix

Mk−1ob,n

∫Ω

NTo ρ(x,NbT

k−1b,n + NoT

k−1o,n )c(x,NbT

k−1b,n +

NoTk−1o,n )Nb dΩ


M′k−1ob,n

∫Ω

NTo

dρdT

∣∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +


k−1b,n + NoT

k−1o,n )NbdΩ


M′k−1ob,n

∫Ω

NTo

dcdT

∣∣(NbT

k−1b,n +NoT

k−1o,n )

(Nb∆Tk−1b,n +


k−1b,n + NoT

k−1o,n )NbdΩ


∫Ω

NTb f(x) dΩ


Fo

∫Ω

NTo f(x) dΩ

Table 4.15: Finite element matrices for nonlinear transient heat equation approximated by Newton-Gauss-Seidel procedure

Method 2: m-step Gauss-Seidel-Newton approach

The derivation of this method starts in the same way as above. First, the Backward-Eulerdiscretization is applied on the PDE level. It is necessary to state, that the nonlinear formin the case of the heat equation is linear in the temperature field. Therefore, it can be split.Then, Equation 4.122, using variational calculus, leads to the following weak formulation ofhp− d problem:

Rb,n(To,n, Tb,n) = F (vb)−A (∆To,n + ∆Tb,n, k(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vb)−−A (To,n−1 + Tb,n−1, k(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vb)−− 〈∆To,n + ∆Tb,n, ρ(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n)...

...c(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vb〉!

= 0 ∀vb ∈ Vb

(4.144)


Ro,n(To,n, Tb,n) = F (vo)−A (∆To,n + ∆Tb,n, k(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vo)−−A (To,n−1 + Tb,n−1, k(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vo)−− 〈∆To,n + ∆Tb,n, ρ(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n)...

...c(To,n−1 + ∆To,n + Tb,n−1 + ∆Tb,n), vo〉!

= 0 ∀vo ∈ Vo

(4.145)

The update of the solution from one step to another is done using the following Equation:

Tb,n = Tb,n−1 + ∆Tb,n

To,n = To,n−1 + ∆To,n(4.146)

Application of nonlinear Gauss-Seidel approach to Equations 4.144 and 4.145 results in:

Rb,n(T l−1o,n , T

lb,n) = F (vb)−A (∆T l−1

o,n + ∆T lb,n, k(To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T lb,n), vb)−

−A (To,n−1 + Tb,n−1, k(To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T lb,n), vb)−

− 〈∆T l−1o,n + ∆T lb,n, ρ(To,n−1 + ∆T l−1

o,n + Tb,n−1 + ∆T lb,n)...

...c(To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T lb,n), vb〉

!= 0 ∀vb ∈ Vb

(4.147)

Ro,n(T lo,n, Tlb,n) = F (vo)−A (∆T lo,n + ∆T lb,n, k(To,n−1 + ∆T lo,n + Tb,n−1 + ∆T lb,n), vo)−−A (To,n−1 + Tb,n−1, k(To,n−1 + ∆T lo,n + Tb,n−1 + ∆T lb,n), vo)−− 〈∆T lo,n + ∆T lb,n, ρ(To,n−1 + ∆T lo,n + Tb,n−1 + ∆T lb,n)...

...c(To,n−1 + ∆T lo,n + Tb,n−1 + ∆T lb,n), vo〉!

= 0 ∀vo ∈ Vo

(4.148)

where l is the counter for Gauss-Seidel iterations.

The linearization of Equation 4.147 is done around the increment δTb and Equation 4.148is around δTo. When the high order terms are omitted, the linearization can be written asfollows:

Rb,n(T l−1o,n , T

l,kb,n) = Rb,n(T l−1

o,n , Tl,k−1b,n ) +DRb,n(T l−1

o,n , Tl,k−1b,n )[δTb] (4.149)

For the overlay the linearization results in:

Ro,n(T l,ko,n, Tlb,n) = Ro,n(T l,k−1

o,n , T lb,n) +DRo,n(T l,k−1o,n , T lb,n)[δTo] (4.150)

The Newton approach approximates ∆T ln as follows:


∆T lb,n = ∆T l,k−1b,n + δTb

∆T lo,n = ∆T l,k−1o,n + δTo

(4.151)

where k is the counter of the Newton iterations.

The residual at any point in time and iteration is required to be zero. Therefore, Equations4.149 and 4.150 are rewritten:

DRb,n(T l−1o,n , T

l,k−1b,n )[δTb] = −Rb,n(T l−1

o,n , Tl,k−1b,n ) (4.152)

DRo,n(T l,k−1o,n , T lb,n)[δTo] = −Ro,n(T l,k−1

o,n , T lb,n) (4.153)

where:

Rb,n(T l−1o,n , T

l,k−1b,n ) =

∫Ω

f(x, t)vbdΩ−∫Ω

ρ(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n )...

...c(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n )∆T l−1

o,n + ∆T l,k−1b,n

∆tvbdΩ−

−∫Ω

∂vb∂x

k(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n )∂(Tb,n−1 + To,n−1)

∂xdΩ−

−∫Ω

∂vb∂x


b,n )∂(∆T l−1

o,n + ∆T l,k−1b,n )

∂xdΩ

(4.154)

Ro,n(T l,k−1o,n , T lb,n) =

∫Ω

f(x, t)vodΩ−∫Ω

ρ(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n)...

...c(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n)

∆T l,k−1o,n + ∆T lb,n

∆tvodΩ−

−∫Ω

∂vo∂x

k(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n)

∂(Tb,n−1 + To,n−1)

∂xdΩ−

−∫Ω

∂vo∂x


∂(∆T l,k−1o,n + ∆T lb,n)

∂xdΩ

(4.155)

The Jacobian matrix of the base and overlay is derived using the product rule of directionalderivative:


DRb,n(T l−1o,n , T

l,k−1b,n )[δTb] =

∫Ω

:

0D(f(x, t)vb)[δTb] dΩ−

−∫Ω

:0D

(∂vb∂x

)[δTb]k(x, To,n−1 + ∆T l−1

o,n + Tb,n−1 + ∆T l,k−1b,n )

∂(Tb,n−1 + To,n−1)

∂xdΩ

−∫Ω

∂vb∂x

D(k(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n ))[δTb]∂(Tb,n−1 + To,n−1)

∂xdΩ

−∫Ω

∂vb∂x


b,n )

:0

D

(∂(Tb,n−1 + To,n−1)

∂x

)[δTb]dΩ

−∫Ω

:0D

(∂vb∂x

)[δTb]k(x, To,n−1 + ∆T l−1

o,n + Tb,n−1 + ∆T l,k−1b,n )

∂(∆T l−1o,n + ∆T l,k−1

b,n )

∂xdΩ

−∫Ω

∂vb∂x


b,n ))[δTb]∂(∆T l−1

o,n + ∆T l,k−1b,n )

∂xdΩ

−∫Ω

∂vb∂x


b,n )D

(∂(∆T l−1

o,n + ∆T l,k−1b,n )

∂x

)[δTb]dΩ

−∫Ω

D(ρ(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n ))[δTb]...


b,n )∆T l−1


∆tvbdΩ−

−∫Ω


b,n )...

...D(c(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n ))[δTb]∆T l−1


∆tvbdΩ−

−∫Ω


b,n )...


b,n )D

(∆T l−1


∆t

)[δTb]vbdΩ−

−∫Ω


b,n )...


b,n )∆T l−1


∆t :

0D(vb)[δTb]dΩ (4.156)

Application of the directional derivative to the overlay residual results in:


DRo,n(T l,k−1o,n , T lb,n)[δTo] =

∫Ω

:0D(f(x, t)vo)[δTo] dΩ−

−∫Ω

:0D

(∂vo∂x

)[δTo]k(x, To,n−1 + ∆T l,k−1

o,n + Tb,n−1 + ∆T lb,n)∂(Tb,n−1 + To,n−1)

∂xdΩ

−∫Ω

∂vo∂x

D(k(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo]

∂(Tb,n−1 + To,n−1)

∂xdΩ

−∫Ω

∂vo∂x


:0

D

(∂(Tb,n−1 + To,n−1)

∂x

)[δTo]dΩ

−∫Ω

:0D

(∂vo∂x

)[δTb]k(x, To,n−1 + ∆T l,k−1

o,n + Tb,n−1 + ∆T lb,n)∂(∆T l,k−1

o,n + ∆T lb,n)

∂xdΩ

−∫Ω

∂vo∂x

D(k(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo]

∂(∆T l,k−1o,n + ∆T lb,n)

∂xdΩ

−∫Ω

∂vo∂x

k(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n)D

(∂(∆T l,k−1

o,n + ∆T lb,n)

∂x

)[δTo]dΩ

−∫Ω

D(ρ(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo]...



∆tvodΩ−

−∫Ω


...D(c(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo]


∆tvodΩ−

−∫Ω


...c(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n)D

(∆T l,k−1

o,n + ∆T lb,n∆t

)[δTo]vodΩ−

−∫Ω




∆t :

0D(vo)[δTo]dΩ (4.157)


The directional derivatives of the terms are expressed as following:


b,n ))[δTb] =

=d

dε

[k(x, To,n−1 + ∆T l−1

o,n + Tb,n−1 + ∆T l,k−1b,n + εδTb)

]ε=0

=

=dk

dTδTb

(4.158)

D(c(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n ))[δTb] =

=d

dε

[c(x, To,n−1 + ∆T l−1


]ε=0

=

=dc

dTδTb

(4.159)

D(ρ(x, To,n−1 + ∆T l−1o,n + Tb,n−1 + ∆T l,k−1

b,n ))[δTb] =

=d

dε

[ρ(x, To,n−1 + ∆T l−1


]ε=0

=

=dρ

dTδTb

(4.160)

D(k(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo] =

=d

dε

[k(x, To,n−1 + ∆T l,k−1

o,n + εδTo + Tb,n−1 + ∆T lb,n)]ε=0

=

=dk

dTδTo

(4.161)

D(c(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo] =

=d

dε

[c(x, To,n−1 + ∆T l,k−1

o,n + εδTo + Tb,n−1 + ∆T lb,n)]ε=0

=

=dc

dTδTo

(4.162)


D(ρ(x, To,n−1 + ∆T l,k−1o,n + Tb,n−1 + ∆T lb,n))[δTo] =

=d

dε

[ρ(x, To,n−1 + ∆T l,k−1

o,n εδTo + Tb,n−1 + ∆T lb,n)]ε=0

=

=dρ

dTδTo

(4.163)

D

(∂(∆T l−1

o,n + ∆T l,k−1b,n )

∂x

)[δTb] =

d

dε

[∂(∆T l−1

o,n + ∆T l,k−1b,n + εδTb)

∂x

]ε=0

=∂(δTb)

∂x(4.164)

D

(∂(∆T l−1

o,n + ∆T l,k−1b,n )

∆t

)[δTb] =

d

dε

[∂(∆T l−1

o,n + ∆T l,k−1b,n + εδTb)

∆t

]ε=0

=δTb∆t

(4.165)

D

(∂(∆T l,k−1

o,n + ∆T lb,n)

∂x

)[δTo] =

d

dε

[∂(∆T l,k−1

o,n + εδTo + ∆T lb,n)

∂x

]ε=0

=∂(δTo)

∂x(4.166)

D

(∂(∆T l,k−1

o,n + ∆T lb,n)

∆t

)[δTo] =

d

dε

[∂(∆T l,k−1

o,n + εδTo + ∆T lb,n)

∆t

]ε=0

=δTo∆t

(4.167)

Equations 4.152 and 4.153 after discretization are written in the compact notation:

Jbb,n(Tl,k−1b ,Tl−1

o )δTb = Fb,n(Tl,k−1b ,Tl−1

o ) (4.168)

Joo,n(Tlb,T

l,k−1o )δTo = Fo,n(Tl

b,Tl,k−1o ) (4.169)

The matrices needed to solve construct Equations 4.168 and 4.169 are shown in Table 4.16.



Base Jacobianmatrix

Jbb,n(Tl,k−1

b ,Tl−1o )

1∆tb

Mbb,n(Tl,k−1b ,Tl−1

o ) + 1∆tb

M′bb,n(Tl,k−1

b ,Tl−1o ) +

1∆tb

M′bb,n(Tl,k−1

b ,Tl−1o ) +(

Kbb,n(Tl,k−1b ,Tl−1

o ) + K′bb,n(Tl,k−1b ,Tl−1

o ))


Joo,n(Tl

b,Tl,k−1o )

1∆to

Moo,n(Tlb,T

l,k−1o ) + 1

∆toM′

oo,n(Tlb,T

l,k−1o ) +

1∆to

M′oo,n(Tl

b,Tl,k−1o ) +(

Koo,n(Tlb,T

l,k−1o ) + K′oo,n(Tl

b,Tl,k−1o )

)Base residual vector Fb,n

(Tl,k−1b ,Tl−1

o )

Fb −Kbb,n(Tl,k−1b ,Tl−1

o )(Tb,n−1 + ∆Tl,k−1

b,n

)−

Kbo,n(Tl,k−1b ,Tl−1

o )(To,n−1 + ∆Tl−1

o,n

)−

1∆tb

Mbb,n(Tl,k−1b ,Tl−1

o )∆Tl,k−1b,n −

1∆to

Mbo,n(Tl,k−1b ,Tl−1

o )∆Tl−1o,n


Fo,n

(Tlb,T

l,k−1o )

Fo −Kob,n(Tlb,T

l,k−1o )

(Tb,n−1 + ∆Tl

b,n

)−

Koo,n(Tlb,T

l,k−1o )

(To,n−1 + ∆Tl,k−1

o,n

)−

1∆tb

Mob,n(Tlb,T

l,k−1o )∆Tl

b,n −1

∆toMoo,n(Tl

b,Tl,k−1o )∆Tl,k−1

o,n

Base stiffness matrix Kbb,n

∫Ω

BTb k(x,NbT

l,k−1b,n + NoT

l−1o,n )Bb dΩ


K′bb,n∫Ω

BTb

dkdT

∣∣(NbT

l,k−1b,n +NoT

l−1o,n )

(BbTl,k−1b,n +

BoTl−1o,n )NbdΩ


Kbo,n

∫Ω

BTb k(x,NbT

l,k−1b,n + NoT

l−1o,n )Bo dΩ


Koo,n

∫Ω

BTo k(x,NbT

lb,n + NoT

l,k−1o,n )Bo dΩ


K′oo,n∫Ω

BTo

dkdT

∣∣(NbT

lb,n+NoT

l,k−1o,n )

(BbTlb,n +

BoTl,k−1o,n )NodΩ


Kob,n

∫Ω

BTo k(x,NbT

lb,n + NoT

l,k−1o,n )Bb dΩ

Base mass matrix Mbb,n

∫Ω

NTb ρ(x,NbT

l,k−1b,n + NoT

l−1o,n ) c(x,NbT

l,k−1b,n +

NoTl−1o,n )Nb dΩ


M′bb,n

∫Ω

NTb

dρdT

∣∣∣(NbT

l,k−1b,n +NoT

l−1o,n )

c(x,NbTl,k−1b,n +

NoTl−1o,n )(Nb∆Tl,k−1

b,n + No∆Tl−1o,n )NbdΩ


M′bb,n

∫Ω

NTb

dcdT

∣∣(NbT

l,k−1b,n +NoT

l−1o,n )

ρ(x,NbTl,k−1b,n +

NoTl−1o,n )(Nb∆Tl,k−1


Coupling massmatrix

Mbo,n

∫Ω

NTb ρ(x,NbT

l,k−1b,n + NoT

l−1o,n )c(x,NbT

l,k−1b,n +

NoTl−1o,n )No dΩ

Overlay mass matrix Moo,n

∫Ω

NTo ρ(x,NbT

lb,n + NoT

l,k−1o,n )c(x,NbT

lb,n +

NoTl,k−1o,n )No dΩ



M′oo,n

∫Ω

NTo

dρdT

∣∣∣(NbT

lb,n+NoT

l,k−1o,n )

(Nb∆Tlb,n +

No∆Tl,k−1o,n )c(x,NbT

lb,n + NoT

l,k−1o,n )NodΩ


M′oo,n

∫Ω

NTo

dcdT

∣∣(NbT

lb,n+NoT

l,k−1o,n )

(Nb∆Tlb,n +

No∆Tl,k−1o,n )ρ(x,NbT

lb,n + NoT

l,k−1o,n )NodΩ

Coupling massmatrix

Mob,n

∫Ω

NTo ρ(x,NbT

lb,n + NoT

l,k−1o,n )c(x,NbT

lb,n +

NoTl,k−1o,n )Nb dΩ


∫Ω

NTb f(x) dΩ


Fo

∫Ω

NTo f(x) dΩ

Table 4.16: Finite element matrices for nonlinear transient heat equation approximated by Gauss-Seidel-Newton procedure

Method 3: one-step Gauss-Seidel-Newton approach

The concept of one-step Gauss-Seidel-Newton approach compared to the one described aboveis the reduction of the inner Newton iteration loop to one. As the formulas for derivationsare complex, they are not presented here. The Equations needed to be solved are as follows:

Jbb,n(Tl−1b ,Tl−1

o )δTb = Fb,n(Tl−1b ,Tl−1

o ) (4.170)

Joo,n(Tlb,T

l−1o )δTo = Fo,n(Tl

b,Tl−1o ) (4.171)

The components for Equations 4.170 and 4.171 are presented in Table 4.17.



Base Jacobianmatrix

Jbb,n(Tl−1

b ,Tl−1o )

1∆tb

Mbb,n(Tl−1b ,Tl−1

o ) + 1∆tb

M′bb,n(Tl−1

b ,Tl−1o ) +

1∆tb

M′bb,n(Tl−1

b ,Tl−1o ) +(

Kbb,n(Tl−1b ,Tl−1

o ) + K′bb,n(Tl−1b ,Tl−1

o ))


Joo,n(Tl

b,Tl−1o )

1∆to

Moo,n(Tlb,T

l−1o ) + 1

∆toM′

oo,n(Tlb,T

l−1o ) +

1∆to

M′oo,n(Tl

b,Tl−1o ) +(

Koo,n(Tlb,T

l−1o ) + K′oo,n(Tl

b,Tl−1o )

)Base residual vector Fb,n

(Tl−1b ,Tl−1

o )Fb −Kbb,n(Tl−1

b ,Tl−1o )

(Tb,n−1 + ∆Tl−1

b,n

)−

Kbo,n(Tl−1b ,Tl−1

o )(To,n−1 + ∆Tl−1

o,n

)−

1∆tb

Mbb,n(Tl−1b ,Tl−1

o )∆Tl−1b,n −

1∆to

Mbo,n(Tl−1b ,Tl−1

o )∆Tl−1o,n


Fo,n

(Tlb,T

l−1o )

Fo −Kob,n(Tlb,T

l−1o )

(Tb,n−1 + ∆Tl

b,n

)−

Koo,n(Tlb,T

l−1o )

(To,n−1 + ∆Tl−1

o,n

)−

1∆tb

Mob,n(Tlb,T

l−1o )∆Tl

b,n −1

∆toMoo,n(Tl

b,Tl−1o )∆Tl−1

o,n

Base stiffness matrix Kbb,n

∫Ω

BTb k(x,NbT

l−1b,n + NoT

l−1o,n )Bb dΩ


K′bb,n∫Ω

BTb

dkdT

∣∣(NbT

l−1b,n +NoT

l−1o,n )

(BbTl−1b,n +BoT

l−1o,n )NbdΩ


Kbo,n

∫Ω

BTb k(x,NbT

l−1b,n + NoT

l−1o,n )Bo dΩ


Koo,n

∫Ω

BTo k(x,NbT

lb,n + NoT

l−1o,n )Bo dΩ


K′oo,n∫Ω

BTo

dkdT

∣∣(NbT

lb,n+NoT

l−1o,n )

(BbTlb,n + BoT

l−1o,n )NodΩ


Kob,n

∫Ω

BTo k(x,NbT

lb,n + NoT

l−1o,n )Bb dΩ

Base mass matrix Mbb,n

∫Ω

NTb ρ(x,NbT

l−1b,n + NoT

l−1o,n ) c(x,NbT

l−1b,n +

NoTl−1o,n )Nb dΩ


M′bb,n

∫Ω

NTb

dρdT

∣∣∣(NbT

l−1b,n +NoT

l−1o,n )

c(x,NbTl−1b,n +

NoTl−1o,n )(Nb∆Tl−1



M′bb,n

∫Ω

NTb

dcdT

∣∣(NbT

l−1b,n +NoT

l−1o,n )

ρ(x,NbTl−1b,n +

NoTl−1o,n )(Nb∆Tl−1


Coupling massmatrix

Mbo,n

∫Ω

NTb ρ(x,NbT

l−1b,n + NoT

l−1o,n )c(x,NbT

l−1b,n +

NoTl−1o,n )No dΩ

Overlay mass matrix Moo,n

∫Ω

NTo ρ(x,NbT

lb,n + NoT

l−1o,n )c(x,NbT

lb,n +

NoTl−1o,n )No dΩ


M′oo,n

∫Ω

NTo

dρdT

∣∣∣(NbT

lb,n+NoT

l−1o,n )

(Nb∆Tlb,n +

No∆Tl−1o,n )c(x,NbT

lb,n + NoT

l−1o,n )NodΩ



M′oo,n

∫Ω

NTo

dcdT

∣∣(NbT

lb,n+NoT

l−1o,n )

(Nb∆Tlb,n +

No∆Tl−1o,n )ρ(x,NbT

lb,n + NoT

l−1o,n )NodΩ

Coupling massmatrix

Mob,n

∫Ω

NTo ρ(x,NbT

lb,n + NoT

l−1o,n )c(x,NbT

lb,n +

NoTl−1o,n )Nb dΩ


∫Ω

NTb f(x) dΩ


Fo

∫Ω

NTo f(x) dΩ

Table 4.17: Finite element matrices for nonlinear transient heat equation approximated by one-stepGauss-Seidel-Newton procedure

4.4.2 Time discretization of the overlay approximation

One of the main advantages of the partitioned approach is the possibility to introduce thesubstepping for the overlay zone to capture the high gradient effects with the better quality.For this case the discretization with different time step for the base and overlay problems isneeded. It was already explained in Section 4.2.2, what is the concept of the overlay timesubstepping and the methods, which can be used for this implementation. It was proven,that if the overlay substepping is performed on the overlay solution only, which is boundedto oscillated around the base approximation, the quality of the final solution is drasticallydecreased compared to the one obtained with the finer time step. For the nonlinear transientcase, only the full solution approximation Thp−d is used to perform the substepping (it wasintroduced as Method 2 in Section 4.2.2).

In the sections above, a different time step for base and overlay problems have been alreadyintroduced. The base solution is discretized with the step ∆tb and the overlay with the step∆to. Moreover, it is necessary to note that, when Method 2 is used, the time step for thebase should be varied withing the overlay zone, i.e. when the base element is overlayed, timestep ∆to is used instead of ∆tb. It is important, as the solution within the performed overlaytime substeps has also the change in the base mesh nodes within the overlay domain.

4.4.3 Implementation of hp− d method

The concept of all methods is similar to the ones described in Section 4.3.2. Therefore, onlythe algorithm for the first method (see Algorithm 7) is described, introducing the possibilityof the overlay substepping. Moreover, the mesh can be moved within specified number oftime steps, thus, the overlay mesh is created withing the time loop.


Algorithm 7 hp-d algorithm for nonlinear transient heat problem with linear elements(Method 1)

1: Create base mesh2: Set Tnb

b and Tnoo to initial condition

3: for nb(iBaseT imeStep) = 1...NumberOfBaseT imeSteps do4: Create/Move overlay mesh5: Set ∆Ti

b and ∆Tio to zero

6: if Substepping = true7: Create overlay transient problem8: for no(iOverlayT imeStep) = 1...NumberOfOverlayT imeSteps do9: for k(iNewton) = 1 ..maximumNumberOfIterations do

10: Solve Jk−1n δTno+1 =

(Fk−1no

)11: Update ∆Tk

no= ∆Tk−1

no+ δTno+1

12: Check convergence of Newton iterations13: if converged14: Set Tno = Tno−1 + ∆Tk

no

15: break Newton Iterations16: end for17: Prepare solution for the coupling18: end if19: Set current base time20: for k(iNewton) = 1 ..maximumNumberOfIterations do21: Set δTl−1

b and δTl−1o to zero

22: for l(iGaussSeidel) = 1 ..maximumNumberOfIterations do23: Compute base solution increment:24: Compute Kk−1

bb,n , K′k−1bb,n , Mk−1

bb,n , M′k−1bb,n , M′k−1

bb,n , Fb

25: Compute coupling terms26: Solve Equation 4.14227: Compute overlay solution increment:28: Compute Kk−1

oo,n, K′k−1oo,n , Mk−1

oo,n, M′k−1oo,n , M′k−1

oo,n , Fo

29: Compute coupling terms30: Solve Equation 4.14331: Check convergence of L2-norm of complete solution32: if L2-norm is converged33: Set ∆Tk

b,n = Tk−1b,n + δTl

b, ∆Tko,n = Tk−1

o,n + δTlo

34: break Gauss-Seidel Iterations35: end if36: Check convergence of Newton iterations37: if converged38: Set Tb,n = Tb,n−1 + ∆Tk

b,n, To,n = To,n−1 + ∆Tko,n

39: break Newton Iterations40: end if41: end for42: end for43: end for44: Postprocess the solution


4.4.4 Verification

Comparison with multilevel hp− solution

Three test cases with increasing complexity are considered in order to verify the code createdin Matlab. The numerical solutions are compared at the last time step.

Test case 1 has linear distribution of thermal diffusivity, density and constant heat capacity:

k(x, T ) = 5T + 1 (4.172)

ρ(x, T ) = 4T + 1 (4.173)

c = 5. (4.174)

The analytical solution for this case is considered to be as follows:

T (x) = −(12 + 3t)x2 + (12 + 3t)x. (4.175)

This setup is tested for fully overlayed ten base elements with ten base time steps and totalsimulation time of t = 0.1. In a second simulation, the partial overlay within x ∈ [0.3; 0.7]is kept constant through the whole simulation time. Further, the moving overlay is tested,keeping it constant within x ∈ [0.1; 0.5] in time t ≤ 0.05s and then moving it to x ∈ [0.3; 0.7].

Test case 2 has the same analytical solution as stated in Equation 4.175, but with sinu-soidal material parameters and corresponding source term. The material for this test case isconsidered as follows:

k(x, T ) = sin(T ) + 10 (4.176)

ρ(x, T ) = cos(T ) + 10 (4.177)

c = sin(T ). (4.178)

This test case is tested with the same parameters as the one above.

Lastly, Gaussian bell distribution of the analytical temperature field is considered:

f(x, t) = exp

(−(x− 0.2− t)2

2 · 0.022

). (4.179)

For this test case, the material parameters are stated below:

k(x, T ) = sin(T ) + 10 (4.180)

ρ(x, T ) = cos(T ) + 10 (4.181)

c = sin(T ) + 10. (4.182)

For this test case all the simulations are performed within the simulation time of t = 0.1


with ten time steps. Moreover, the moving overlay is considered within x ∈ [0.0; 0.3] whentime t ≤ 0.05 and then moved to x ∈ [0.1; 0.4]. The base discretization consists of forty baseelements.

The results for the full overlay according to the used method is shown in Table 4.18, forpartial overlay in Table 4.19 and for moving overlay in Table 4.20. The number of necessaryiterations is marked in average per time step. It is shown, that all of the test cases gives onlynumerical difference in comparison to the monolithic multilevel hp− solution.

Case Method 1 Method 2 Method 3

1

e = 1.6184 · 10−30 e = 2.2847 · 10−26 e = 4.2387 · 10−26

3 Newton iterations 13 GS iterations

13 merged iterations13 GS (each)

1: 3 and 2 Newton;2-4: 2 and 2; 5-13: 1

and 1

2

e = 1.0498 · 10−30 e = 4.9583 · 10−30 e = 4.6162 · 10−29


7 merged iterations8 GS (each)1: 3 and 2 Newton; 2:2 and 2; 3-8: 1 and 1

3

e = 1.3070 · 10−29 e = 9.4735 · 10−24 e = 5.0653 · 10−22


7 merged iterations1: 7 GS; 2: 8 GS; 3-4:9 GS

1: 4 and 3 Newton; 2:3 and 2; 3: 2 and 2;

4-7: 1 and 1

Table 4.18: Verification of the multiscale code with the multilevel hp-code for nonlinear transienttest cases considering full overlay



1


e = 1.6817 · 10−30 e = 7.3700 · 10−27 e = 7.1577 · 10−27

3 Newton iterations 13 GS iterations13 mergediterations13 GS (each)

1: 3 and 2 Newton;2-4: 2 and 2; 5-13: 1

and 1

2


e = 1.9461 · 10−30 e = 1.1476 · 10−29 e = 1.4213 · 10−29


1: 3 and 2 Newton; 2:2 and 2; 3-6: 1 and 1

3


e = 8.2555 · 10−30 e = 1.4083 · 10−23 e = 3.9505 · 10−22

4 Newton iterations 7 GS iterations7 mergediterations1: 7 GS; 2: 8 GS; 3-4:

9 GS


4-7: 1 and 1

Table 4.19: Verification of the multiscale code with the multilevel hp-code for nonlinear transienttest cases considering partial overlay


1

Partial overlay: Ωo = [0.1, 0.3]→ [0.3, 0.7]

e = 1.1432 · 10−30 e = 8.7398 · 10−27 e = 1.0766 · 10−26


1: 3 and 2 Newton;2-4: 2 and 2; 5-13: 1

and 1

2


e = 1.3041 · 10−30 e = 2.5582 · 10−28 e = 2.0859 · 10−28


1: 3 and 2 Newton; 2:2 and 2; 3-6: 1 and 1

3


e = 4.8796 · 10−30 e = 3.0839 · 10−23 e = 5.0428 · 10−22

4 Newton iterations 7 GS iterations7 mergediterations1: 7 GS; 2: 8 GS; 3-4:

9 GS


4-7: 1 and 1

Table 4.20: Verification of the multiscale code with the multilevel hp-code for nonlinear transienttest cases considering moving overlay



The multi-level hp−method does not allow to use different time steps for the base and overlaymeshes, while multiscale hp − d scheme allows to introduce the substepping for the overlayapproximation. The aim of this thesis is to show, that the substepping improves the qualityof the result while decreasing the numerical costs: the whole system does not need to besolved with the finer time step.

Test case 2 is tested with the ten base elements, each overlayed with two, within the simulationtime of t = 0.1 s. The overlay time step is kept the same at the level of ∆to = 0.01 s. Thebase time step is constantly increased. The multiscale hp − d solution is compared to themonolithic multi-level hp− solution with the time step of ∆t = 0.01 s and the same physicalsetup.

Table D.1 shows that, in case of the full overlay, the increase of the base time step, keeping theoverlay time step the same, does not have an influence on the numerical solution: the com-parison with the monolithic approach shows only numerical difference. When the base timestep is increased, while analyzing the partial overlay, the quality of the solution, comparedto the monolithic approach drastically decreases. There are two reasons to be noted.

First, the multiscale hp− d approach implies the introduction of the new sub time step onlywithin the overlay zone. The rest of the domain solution is kept the same from one base stepto another. The principal difference of the multilevel hp− approach, when the time is refined,is that the whole domain is solved within each time step. Therefore, the solution outside ofthe overlay zone is also affected by the new time steps. “Freezing” the base solution outside,while advancing in time for the overlay domain, leads to the significant errors, especially ifthe solution is highly changing within this zone.

The second reason for the appeared difference is in the application of the initial condition.In the multilevel hp− code in AdhoC++, the initial condition is calculated with the helpof L2−projection, obtained from the non-constrained unit mass matrix and the force vec-tor, calculated based on the continuous initial condition function. It was checked, that theprojection led to the non-zero values at the boundaries of the bar, while the homogeneousDirichlet boundary condition is prescribed. When the first time step solution is calculated,the constraining of the matrices takes into consideration the projection and sets the valuesat the boundaries to the correct prescribed ones. The multiscale hp− d approach follows thesame procedure in case no overlay substepping is introduced. When the first overlay substepis introduced and the partial overlay is considered, the boundary conditions to the problemare prescribed differently. The nonhomogeneous Dirichlet boundary conditions are appliedto the overlay region with the fixed values of the base time step solution. This is necessary inorder to avoid the discontinuity of the solution field over the border between base and overlayzones, while performing the coupling time step. When the substepping is performed beforethe first coupling step, the boundary conditions values are taken from the L2 projection of theinitial condition and kept the same during the overlay substeps. When the coupling is per-formed, the base domain is constrained according to the prescribed homogeneous Dirichlet.This influences the quality of the obtained solution. For this reason, the principal approachshould be changed: the initial L2 projection should be performed for both cases based on theconstrained unit mass matrix and the source vector, then the prescribed constrains will beautomatically fulfilled.

Test case 3 is considered with forty base elements, but with the smaller time step in order to


capture the physical solution. For this case the multilevel hp−solution is generated with 100time steps (∆t = 0.001 s) and the total simulation time of t = 0.1 s. For the full overlay onlynumerical error is obtained between multiscale and multilevel solutions (see Table D.2). Whenthe partial overlay is considered, the same drop of the quality of the solution is observed,increasing the base time step. This is due to the reasons explained above. It is interesting tonote that, when Method 2 or 3 are used for partial and overlay cases with no substepping, thesolution quality drops by approximately 6 orders of magnitudes (see full overlay for Method 1and 2, for example, in Table D.2). It can indicate either the numerical zero of two solutions,the conceptual difference in the considered approaches, or the implementational error. Asthe increase of the quality of the solution is introduced with the increase of the used subtime steps, the drop is, mostly, happening because of the difference in the methodologicalapproaches. As no such difference was observed before in nonlinear stationary case andfor more simple test cases in nonlinear transient simulations, it is suggested to investigatethe performance of the Gauss-Seidel-Newton methods when applied to nonlinear transientproblems.

Convergence studies

The convergence studies are performed considering test case 2 and 3. The analytical solutionsfor these cases was shown in Equation 4.175 and 4.179 respectively. For test case 2 it wasobserved, that with 10 base time steps and the total simulation time of t = 0.1 s the relativeerror stays at the same level. Therefore, only refinement in space is performed, increasingthe number of overlayed elements for each of the ten base elements. Figure 4.29 and TableD.3 show that all of the methods have the same convergence in space as expected. Moreover,the difference between methods in the relative error is about 10−19.

10 20 30 40 50 60 70 80 90 100

10−12

10−11

10−10

10−9

10−8

10−7

10−6

Number of overlay elements per each base element

L2

erro

r,[

-]

Full overlay method 1Full overlay method 2Full overlay method 3



The studies of the influence of the time step are performed for test case 3. The setup consistsof forty base elements, each of them overlayed with two elements. The sub time steps wereintroduced per base time step with the total simulation time of t = 0.1 s. Table D.4 andFigure 4.30 show that, at about 20 sub time steps, the error of the numerical solution canbe considered to be leveled off. Therefore, this configuration will be further refined in space.The difference between the relative errors from different methods is about 10−18. The sameconclusion holds for all of the studied methods: the rate of convergence in time for all ofthem is the same.

10 20 30 40 50 60 70 80 90 100

10−3

10−2.9

10−2.8

10−2.7

Number of time steps per base time step

L2

erro

r,[

-]


Figure 4.30: Convergence studies on time refinement for Test case 3 considering 40 base elements,each overlayed with 2

Table D.5 and Figure 4.31 present the convergence results on the spatial refinement for testcase 3 considering forty base elements, ten base time steps and 20 overlay sub steps. Thenumber of overlay elements per base element is increased from 2 to 100. As the consideredtest case is hard to capture the physical effects, the convergence studies should be continuedfurther, considering each level off in time and space. The studies were held till the set up offorty base elements with overlay of thirty elements per each, considering 10 base time stepswith 90 sub time steps each. The relative L2-error drop for this setup till 3.5739 · 10−7. Theconvergence analysis again has shown that all of the methods are equivalent in the rate ofconvergence in time and space.


10 20 30 40 50 60 70 80 90 100

10−5

10−4

10−3

Number of overlay elements per each base element

L2

erro

r,[

-]


Figure 4.31: Convergence studies on space refinement for Test case 3 considering 10 base time stepsand 20 overlay sub steps

Costs studies

In order to give a first estimate on the performance of all three methods for the nonlineartransient case, test case 2 and 3 with the full overlay are tested. The outer loop for Method 1is the Newton iteration loop, for which the convergence criteria is the inner product betweenthe increment of the solution and the residual. For Method 2 as the Gauss-Seidel is the outeriteration loop, the relative L2-error of the complete solution is recorded after each iteration.The same holds for method 3, where only one merged Gauss-Seidel-Newton iteration loop isconsidered. All of the evaluations are made in the last time step.

Figure 4.32 shows the number of the respective calls of the “\”against the relative error asthe convergence criteria for test case 2. The number of the calls is already high, as ten basetime steps are already performed. It is proven that, Method 3 needs much more iterationsto converge. Method 1 needs the most computational costs to reach the convergent solution.

Test case 3 is considered with forty base elements, each overlayed with two. The convergencecriteria is recorded at the last time step. Figure 4.33 shows that Method 3 performs betterthan Method 2. Therefore, based on the performed studies, Method 3 is recommended asthe best balance between implmentational and computational costs.


102 102.1 102.2 102.3 102.4 102.5 102.610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Number of the calls of ”\”

Rel

ativ

eL

2er

ror,

[-

]

Method 1Method 2Method 3

Figure 4.32: Number of the calls of “\” against the convergence criteria for Test case 2 with fulloverlay, recorded at the last time step

102 102.1 102.2 102.3 102.4 102.5 102.6 102.7 102.810−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Number of the calls of ”\”

Rel

ativ

eL

2er

ror,

[-

]

Method 1Method 2Method 3

Figure 4.33: Number of the calls of “\” against the convergence criteria for Test case 3 with fulloverlay, recorded at the last time step


4.4.5 Significance of the precision of the Jacobian computation and thecoupling terms

The precision of the computations of the Jacobian is depending not only on the presence ofthe derivative of the thermal diffusivity, but also the derivatives of the heat capacity anddensity. First, the derivative of stiffness term will be neglected, then the derivative of themass terms. Test for this purpose are performed on test case 2 with ten base elements andfull overlay of two elements per each. The total simulation time is t = 0.1 s with 10 timesteps.

Significance of the precision of the Jacobian computation for Method 1, 2 and 3 can be seenin Table D.6, D.8 and D.10 respectively. As before, the method needs more iterations toconverge, when the Jacobian is computed not precisely, but it does not have an influence onthe relative error of the numerical solution compared to the analytical.

The importance of the coupling terms is estimated as the relative input of the current termto the right-hand side of the equation. The formulas, used for Method 1, 2 and 3, are asfollows:

SoK =‖Kbo,n

(To,n−1 + ∆Tl−1

o,n

)‖L2

‖Fb‖L2

(4.183)

SbK =‖Kob,n

(Tb,n−1 + ∆Tl

b,n

)‖L2

‖Fo‖L2

(4.184)

SoM =‖ 1

∆tbMbb,n∆Tl−1

b,n ‖L2

‖Fb‖L2

(4.185)

SbM =‖ 1

∆tbMob,n(Tl

b,Tl−1o )∆Tl

b,n‖L2

‖Fo‖L2

(4.186)

For method 1, test case 2 is considered. The same convergence analysis is performed with-out all extra coupling terms, originating from the coupling Jacobian Jk−1

bo,nδTl−1o and Jk−1

ob,nδTlb.

The results of the convergence analysis can be seen in Table D.7. It is visible, that the relativeL2-error, which the solution converge to, is numerically different. The difference between thenumerical solution, based on full coupling and neglected coupling terms, is about 10−11. Thenumber of necessary iterations is drastically decreased. The terms Kbo,n

(To,n−1 + ∆Tl−1

o,n

),

Kob,n

(Tb,n−1 + ∆Tl

b,n

), 1

∆tbMbb,n∆Tl−1

b,n and 1∆tb

Mob,n(Tlb,T

l−1o )∆Tl

b,n can not be neglected,

as their influence on the numerical solution is high.

The test for Methods 2 and 3 was performed considering test case 3 with forty base elements,each overlayed with two. The total simulation time was t = 0.1 s, discretized with ten basetime steps.

In Tables D.9 and D.11 these factors are shown for Method 2 and 3 respectively with respectto the Gauss-Seidel iterations for the last time step. None of the terms can be neglected, asthey have strong influence on the quality of the numerical solution.

107

Chapter 5

Summary and Conclusions

The main aim of this work was to develop a partitioned multiscale hp − d approach, whichcan be applied for the solution of nonlinear transient heat problems. This method allowsto reduce the computational costs for problems with a very local thermal effect, such as theproblems of additive manufacturing. In order to show that application of the partitionedapproach is possible for these physical problems, an one-dimensional models were considered.

First, the linear stationary one-dimensional heat equation was studied. The overlay approx-imation was applied to the governing equations and the partitioned numerical model wasobtained. It was shown that two independent models, overlay and base models, can beconstructed and solved separately, accounting for the volumetric coupling effect of one to an-other. It was demonstrated that the implementation of the coupling terms is simple, as theycan be considered as an extra force term, originating form the prestrain of another model.Attention should be paid to the computation of stiffness matrices and force terms of the baseproblem, as a composed Gauss integration needs to be applied in order to obtain preciseintegration of the system matrices. The considered study cases were verified, using in-househigh-order finite element code AdhoC + +. The multilevel hp−concept, if it is restricted toone overlay level and linear elements on it, is representing the monolithic multiscale hp− d.Therefore, the results from the partitioned method were compared to the results of the mono-lithic model. After verifying the code, the studies on the significance of the coupling termswere performed. It was shown that none of the coupling terms can be neglected to obtain aconvergent solution.

Second, the overlay approximation was applied to the linear transient heat equation. In thescope of this work, a Backward-Euler time integration scheme was applied. The decoupling ofthe base and overlay problems led to the introduction of two extra coupling terms, originatingfrom the inertia effects and from the vatiation of the solution in time. The main interest ofthese studies was in the introduction of the substepping for the overlay region. This candrastically reduce the computational effort in additive manufacturing simulations, as thesolution within a local effect zone can be improved in quality by advancing it in smaller timesteps, without solving the whole system. It was shown that the first method uses the overlaysolution To to perform the substeppping. This solution is restricted to oscillate aroundthe base solution. Accordingly, the complete numerical solution delivers a poor quality.Therefore, the second method, using the complete hp−d solution for the overlay substepping,was developed. It was shown, that this approach leads to the same solution quality compared

108 5. Summary and Conclusions

to the monolithic hp− d formulation, providing with a fine overlay time step, applied for thewhole system. The studies on the significance of the coupling terms revealed that all extracoupling terms are important to obtain a convergent solution.

Further, the nonlinear stationary case was studied. The main challenge in the nonlinearsimulations is the decoupling procedure. As the nonlinear form can not be simply split, moreadvanced methods need to be applied. As the straightforward approach of, first, lineariz-ing, and then splitting the nonlinear problem leads to a significant increase of the couplingterms, which increases the implementational and computational effort, two more methodswere considered: the m-step Gauss-Seidel-Newton and the one-step Gauss-Seidel-Newton.The numerical method for each method was developed, leading to a decoupled system ofbase and overlay equations. After the verification of the implemented code, it was interestingto see how the spatial convergence differs from one method to another. It was shown that allconsidered methods have the same convergence rate in space. Therefore, the convergence rateof the numerical methods itself was compared. If both, numerical and implementational ef-forts are considered, the m-step Gauss-Seidel-Newton and the one-step Gauss-Seidel-Newtonhave shown the best results. Therefore, it is recommended to use these two approaches tofully decouple the nonlinear system. The significance of the extra terms arising from thedecoupling has shown that, for the Newton-Gauss-Seidel approach the terms, involving theincrement in the solution can be neglected independent on the test case. The convergence ofthe solution was achieved. However, the unprecise computation of the Jacobian matrices ledto an increase of the numerical effort, needed to reach the convergent result.

Last, the nonlinear transient case was investigated. The Newton-Gauss-Seidel and Gauss-Seidel-Newton approaches were developed in order to decouple the nonlinear transient systemof equations. The same conclusion as for the nonlinear stationary case was made: the Gauss-Seidel-Newton methods give the same convergence rate in space and time as the Newton-Gauss-Seidel method. Therefore, they are recommended to be used considering implemen-tational and runtime costs. Moreover, the substepping technique within the overlay zoneprovided only a numerical difference in comparison to the multilevel solution obtained witha finer time step. Nevertheless, the same problem with “freezing” the base solution, whilethe overlay solution is advanced in time, was observed. It should be taken into consideration,when the overlay substepping is performed, as it leads to a drastic increase of the quality ofthe solution.

As the present work is opening a whole range of further investigations in the context of thepartitioned multiscale hp− d approach, the following further steps are suggested:

• Extension of the overlay approximation concept to two- and three-dimensional models.This would enable a more detailed study of the numerical and implementational effort.The partitioned approach allows full decoupling of base and overlay system, whichenables an overlay time sub stepping. But at the same time, the communication costsbetween the two meshes increases. The two- and three-dimensional models would allowto estimate this increase and make a conclusion about the efficiency of the presentedapproach. Moreover, the communication structure and the computation of the systemmatrices can be improved. As some of the coupling terms are only dependent on theouter loop solution, they can be computed only once, which will reduce the numericaleffort. These studies are recommended to be held in more dimensional models.

• High-order elements are recommended to be introduced on the base or overlay level.

109

As the problem of the “frozen” base solution for transient cases, while introducing theoverlay sub stepping, leads to a decrease of the quality of the solution, the use of thehigh order Ansatz functions for the base elements can improve these results.

• Introduction of error estimators. The quality of the base mesh is highly influencing thequality of the overlay solution, when the coupling is performed. In order to reduce thenumerical effort and improve the numerical solution, an error estimator, prior to thecoupling between base and overlay meshes should be used.

• Use of high-order time integration schemes. For transient problems, the first-orderBackward-Euler time integration scheme may lead to inaccurate results in manufactur-ing processes. Therefore, it is recommended to implement high-order time integrationschemes from the family of Runge-Kutta methods.

• More overlays can be used. In order to get a very high precision of the gradient withinthe local zones, more overlay levels can be used. For these case, the overlay approxi-mation should be developed with more overlay regions.

• For the case of nonlinear models, more methods can be used in order to decouple thesystem of equations, e.g. m-step SOR-Newton iterations. Moreover, it is interestingto investigate the limitations of the considered approaches for more complex nonlineartransient cases and detect their range of applicability.

It was shown within the current work, that the partitioned hp− d method can be applied tononlinear transient problems. It has benefits, compared to the monolithic approach, such aslower numerical effort together with the higher quality of the numerical solution. Althoughthe work of this thesis is limited to the one-dimensional problems, the gained insights can beused for further development and implementation of the multiscale hp− d method for morecomplex cases.

110 5. Summary and Conclusions

111

Appendix A

Linear stationary heat equation

Influence of coupling terms for the linear stationary heat equation

Number of Full overlay Partial overlayoverlay elements So Sb So Sb

1 0 0 0 0

2 4.8840 · 10−8 1.5005 · 10−6 1.3625 · 10−2 1.1947 · 10−3

3 6.8626 · 10−8 2.1081 · 10−6 1.3626 · 10−2 1.1950 · 10−3

4 7.6346 · 10−8 2.3451 · 10−6 1.3626 · 10−2 1.1951 · 10−3

5 8.0058 · 10−8 2.4591 · 10−6 1.3627 · 10−2 1.1952 · 10−3

10 8.5146 · 10−8 2.6152 · 10−6 1.3627 · 10−2 1.1953 · 10−3

15 8.6105 · 10−8 2.6447 · 10−6 1.3627 · 10−2 1.1953 · 10−3

20 8.6443 · 10−8 2.6550 · 10−6 1.3627 · 10−2 1.1953 · 10−3

25 8.6599 · 10−8 2.6598 · 10−6 1.3627 · 10−2 1.1953 · 10−3

30 8.6684 · 10−8 2.6624 · 10−6 1.3627 · 10−2 1.1953 · 10−3

35 8.6735 · 10−8 2.6640 · 10−6 1.3627 · 10−2 1.1953 · 10−3

40 8.6768 · 10−8 2.6650 · 10−6 1.3627 · 10−2 1.1953 · 10−3

45 8.6791 · 10−8 2.6657 · 10−6 1.3627 · 10−2 1.1953 · 10−3

50 8.6807 · 10−8 2.6662 · 10−6 1.3627 · 10−2 1.1953 · 10−3

100 8.6859 · 10−8 2.6678 · 10−6 1.3627 · 10−2 1.1953 · 10−3

150 8.6869 · 10−8 2.6681 · 10−6 1.3627 · 10−2 1.1953 · 10−3

Table A.1: Influence of the coupling terms for Test Case 2 in linear stationary heat equation

112 A. Linear stationary heat equation

Number of So 6= 0 So 6= 0 So = 0overlay elements Sb 6= 0 Sb = 0 Sb 6= 0

1 1.1377 · 10−4 1.1377 · 10−4 1.1377 · 10−4

2 7.1184 · 10−6 9.9061 · 10−6 8.0815 · 10−6

3 1.4064 · 10−6 3.6434 · 10−6 1.9522 · 10−6

4 4.4503 · 10−7 2.4104 · 10−6 8.0072 · 10−7

5 1.8232 · 10−7 2.0081 · 10−6 4.4227 · 10−7

10 1.1394 · 10−8 1.6374 · 10−6 1.3614 · 10−7

15 2.2506 · 10−9 1.5896 · 10−6 1.0100 · 10−7

20 7.1211 · 10−10 1.5744 · 10−6 9.0297 · 10−8

25 2.9168 · 10−10 1.5676 · 10−6 8.5620 · 10−8

30 1.4066 · 10−10 1.5640 · 10−6 8.3153 · 10−8

35 7.5927 · 10−11 1.5618 · 10−6 8.1691 · 10−8

40 4.4507 · 10−11 1.5605 · 10−6 8.0753 · 10−8

45 2.7786 · 10−11 1.5595 · 10−6 8.0114 · 10−8

50 1.8230 · 10−11 1.5588 · 10−6 7.9659 · 10−8

100 1.1394 · 10−12 1.5567 · 10−6 7.8218 · 10−8

150 2.2506 · 10−13 1.5563 · 10−6 7.7953 · 10−8

Table A.2: Influence of the coupling terms for Test Case 2 on the convergence behaviour in linearstationary heat equation

Number of Full overlay Partial overlayoverlay elements So Sb So Sb

1 0 0 0 0

2 1.6708 · 10−7 1.4116 · 10−7 1.0809 · 10−4 2.8020 · 10−5

3 2.3395 · 10−7 1.9769 · 10−7 1.0658 · 10−4 2.8016 · 10−5

4 2.5995 · 10−7 2.1867 · 10−7 1.0606 · 10−4 2.8073 · 10−5

5 2.7244 · 10−7 2.3023 · 10−7 1.0583 · 10−4 2.8126 · 10−5

10 2.8954 · 10−7 2.4469 · 10−7 1.0553 · 10−4 2.8276 · 10−5

15 2.9276 · 10−7 2.4741 · 10−7 1.0548 · 10−4 2.8339 · 10−5

20 2.9389 · 10−7 2.4837 · 10−7 1.0546 · 10−4 2.8373 · 10−5

25 2.9442 · 10−7 2.4881 · 10−7 1.0545 · 10−4 2.8394 · 10−5

30 2.9470 · 10−7 2.4905 · 10−7 1.0545 · 10−4 2.8409 · 10−5

35 2.9487 · 10−7 2.4920 · 10−7 1.0545 · 10−4 2.8419 · 10−5

40 2.9498 · 10−7 2.4929 · 10−7 1.0545 · 10−4 2.8427 · 10−5

45 2.9506 · 10−7 2.4936 · 10−7 1.0545 · 10−4 2.8433 · 10−5

50 2.9512 · 10−7 2.4940 · 10−7 1.0545 · 10−4 2.8438 · 10−5

100 2.9529 · 10−7 2.4955 · 10−7 1.0545 · 10−4 2.8461 · 10−5

150 2.9532 · 10−7 2.4958 · 10−7 1.0545 · 10−4 2.8469 · 10−5

Table A.3: Influence of the coupling terms for Test Case 3 in linear stationary heat equation

113

Number of So 6= 0 So 6= 0 So = 0overlay elements Sb 6= 0 Sb = 0 Sb 6= 0

1 1.5940 · 10−3 1.5940 · 10−3 1.5940 · 10−3

2 1.0214 · 10−4 1.0324 · 10−4 1.0363 · 10−4

3 2.0265 · 10−5 2.0929 · 10−5 2.1326 · 10−5

4 6.4237 · 10−6 6.8905 · 10−6 7.2839 · 10−6

5 2.6331 · 10−6 3.0006 · 10−6 3.3919 · 10−6

10 1.6342 · 10−7 3.9004 · 10−7 7.7657 · 10−7

15 3.2545 · 10−8 2.3095 · 10−7 6.1774 · 10−7

20 1.0300 · 10−8 1.9946 · 10−7 5.8496 · 10−7

25 4.1973 · 10−9 1.8872 · 10−7 5.7392 · 10−7

30 1.9734 · 10−9 1.8450 · 10−7 5.6964 · 10−7

35 1.1445 · 10−9 1.8184 · 10−7 5.6778 · 10−7

40 6.9034 · 10−10 1.8041 · 10−7 5.6594 · 10−7

45 4.4690 · 10−10 1.7961 · 10−7 5.6540 · 10−7

50 3.1326 · 10−10 1.7922 · 10−7 5.6518 · 10−7

100 1.0406 · 10−10 1.7724 · 10−7 5.6298 · 10−7

150 9.0329 · 10−11 1.7721 · 10−7 5.6259 · 10−7

Table A.4: Influence of the coupling terms for Test Case 3 on the convergence behaviour in linearstationary heat equation

114 A. Linear stationary heat equation

115

Appendix B

Linear transient heat equation

Convergence results of the test cases for transient heat equation

Number of Full overlay Partial overlaysubsteps Method 1 Method 2 Method 1 Method 2

2 1.0546 · 10−3 9.7933 · 10−4 1.0547 · 10−3 9.7930 · 10−4

3 1.0514 · 10−3 9.5290 · 10−4 1.0515 · 10−3 9.5288 · 10−4

4 1.0496 · 10−3 9.4004 · 10−4 1.0497 · 10−3 9.4002 · 10−4

5 1.0483 · 10−3 9.3244 · 10−4 1.0484 · 10−3 9.3242 · 10−4

6 1.0473 · 10−3 9.2742 · 10−4 1.0475 · 10−3 9.2741 · 10−4

7 1.0464 · 10−3 9.2387 · 10−4 1.0466 · 10−3 9.2386 · 10−4

8 1.0456 · 10−3 9.2121 · 10−4 1.0459 · 10−3 9.2120 · 10−4

9 1.0449 · 10−3 9.1916 · 10−4 1.0452 · 10−3 9.1915 · 10−4

10 1.0443 · 10−3 9.1752 · 10−4 1.0446 · 10−3 9.1751 · 10−4

11 1.0436 · 10−3 9.1618 · 10−4 1.0440 · 10−3 9.1617 · 10−4

12 1.0430 · 10−3 9.1507 · 10−4 1.0434 · 10−3 9.1506 · 10−4

15 1.0414 · 10−3 9.1263 · 10−4 1.0419 · 10−3 9.1263 · 10−4

20 1.0389 · 10−3 9.1021 · 10−4 1.0395 · 10−3 9.1020 · 10−4

30 1.0346 · 10−3 9.0779 · 10−4 1.0354 · 10−3 9.0779 · 10−4

40 1.0307 · 10−3 9.0659 · 10−4 1.0318 · 10−3 9.0659 · 10−4

80 1.0184 · 10−3 9.0479 · 10−4 1.0202 · 10−3 9.0479 · 10−4

120 1.0095 · 10−3 9.0420 · 10−4 1.0116 · 10−3 9.0419 · 10−4

160 1.0026 · 10−3 9.0390 · 10−4 1.0051 · 10−3 9.0389 · 10−4

300 9.8768 · 10−4 9.0348 · 10−4 9.9040 · 10−4 9.0347 · 10−4

500 9.7698 · 10−4 9.0329 · 10−4 9.7963 · 10−4 9.0328 · 10−4

Table B.1: Convergence studies on time refinement for Test case 3

116 B. Linear transient heat equation

Overlay per each base element Full overlay Partial overlay

2 9.0348 · 10−4 9.0347 · 10−4

3 1.8401 · 10−4 1.8401 · 10−4

4 5.8905 · 10−5 5.8903 · 10−5

5 2.4278 · 10−5 2.4277 · 10−5

10 1.1758 · 10−5 1.1758 · 10−5

15 1.5440 · 10−6 1.5437 · 10−6

20 1.0097 · 10−7 1.0093 · 10−7

50 3.5685 · 10−9 3.6042 · 10−9

Table B.2: Convergence studies on spatial refinement for Test case 3

Influence of coupling terms for the linear transient heat equation

Number ofSbK SbM SoK SoMsubsteps

2 0 2.0126 · 10−4 3.1751 · 10−30 3.7841 · 10−6

3 0 2.0191 · 10−4 4.8175 · 10−30 3.8646 · 10−6

4 0 2.0218 · 10−4 4.8175 · 10−30 3.8947 · 10−6

5 0 2.0233 · 10−4 5.6714 · 10−30 3.9092 · 10−6

6 0 2.0242 · 10−4 5.6714 · 10−30 3.9173 · 10−6

7 0 2.0249 · 10−4 5.6714 · 10−30 3.9224 · 10−6

8 0 2.0253 · 10−4 5.6714 · 10−30 3.9258 · 10−6

9 0 2.0257 · 10−4 6.0554 · 10−30 3.9282 · 10−6

10 0 2.0259 · 10−4 6.0554 · 10−30 3.9299 · 10−6

11 0 2.0261 · 10−4 6.0554 · 10−30 3.9312 · 10−6

12 0 2.0263 · 10−4 6.0554 · 10−30 3.9322 · 10−6

15 0 2.0267 · 10−4 6.0554 · 10−30 3.9343 · 10−6

20 0 2.0271 · 10−4 6.9077 · 10−30 3.9360 · 10−6

30 0 2.0274 · 10−4 6.9077 · 10−30 3.9374 · 10−6

40 0 2.0276 · 10−4 6.9077 · 10−30 3.9379 · 10−6

80 0 2.0279 · 10−4 6.9077 · 10−30 3.9386 · 10−6

120 0 2.0280 · 10−4 6.9077 · 10−30 3.9388 · 10−6

160 0 2.0280 · 10−4 6.9077 · 10−30 3.9389 · 10−6

300 0 2.0281 · 10−4 6.9077 · 10−30 3.9390 · 10−6

500 0 2.0281 · 10−4 6.9077 · 10−30 3.9391 · 10−6

Table B.3: Influence of the coupling terms for Test Case 3 in linear transient heat equation dependingon the number of the considered time substeps

117

Number ofSbK SbM SoK SoMoverlayed elements

2 0 2.0233 · 10−4 5.6714 · 10−30 3.9092 · 10−6

3 0 1.8847 · 10−4 5.5816 · 10−30 5.7902 · 10−6

4 0 1.8487 · 10−4 5.6804 · 10−30 6.5637 · 10−6

5 0 1.8365 · 10−4 5.7108 · 10−30 6.9430 · 10−6

6 0 1.8323 · 10−4 5.7414 · 10−30 7.1548 · 10−6

10 0 1.8329 · 10−4 5.6571 · 10−30 7.4703 · 10−6

20 0 1.8415 · 10−4 5.5302 · 10−30 7.6060 · 10−6

50 0 1.8501 · 10−4 5.8358 · 10−30 7.6443 · 10−6

Table B.4: Influence of the coupling terms for Test Case 3 in linear transient heat equation dependingon the number of the overlayed elements per each base element

Number ofSbK SbM SoK SoMsubsteps

2 2.5951 · 10−6 1.9103 · 10−6 6.1144 · 10−8 3.5892 · 10−8

3 2.5955 · 10−6 1.9135 · 10−6 6.1153 · 10−8 3.6596 · 10−8

4 2.5957 · 10−6 1.9146 · 10−6 6.1157 · 10−8 3.6849 · 10−8

5 2.5958 · 10−6 1.9151 · 10−6 6.1160 · 10−8 3.6967 · 10−8

6 2.5959 · 10−6 1.9154 · 10−6 6.1162 · 10−8 3.7032 · 10−8

7 2.5959 · 10−6 1.9155 · 10−6 6.1163 · 10−8 3.7070 · 10−8

8 2.5960 · 10−6 1.9156 · 10−6 6.1164 · 10−8 3.7095 · 10−8

9 2.5960 · 10−6 1.9157 · 10−6 6.1165 · 10−8 3.7112 · 10−8

10 2.5960 · 10−6 1.9158 · 10−6 6.1165 · 10−8 3.7124 · 10−8

11 2.5960 · 10−6 1.9158 · 10−6 6.1166 · 10−8 3.7133 · 10−8

12 2.5961 · 10−6 1.9158 · 10−6 6.1166 · 10−8 3.7140 · 10−8

15 2.5961 · 10−6 1.9159 · 10−6 6.1167 · 10−8 3.7153 · 10−8

20 2.5961 · 10−6 1.9160 · 10−6 6.1168 · 10−8 3.7162 · 10−8

30 2.5962 · 10−6 1.9160 · 10−6 6.1169 · 10−8 3.7169 · 10−8

40 2.5962 · 10−6 1.9160 · 10−6 6.1170 · 10−8 3.7171 · 10−8

80 2.5962 · 10−6 1.9160 · 10−6 6.1170 · 10−8 3.7173 · 10−8

120 2.5962 · 10−6 1.9160 · 10−6 6.1171 · 10−8 3.7173 · 10−8

160 2.5962 · 10−6 1.9160 · 10−6 6.1171 · 10−8 3.7173 · 10−8

300 2.5962 · 10−6 1.9160 · 10−6 6.1171 · 10−8 3.7173 · 10−8

500 2.5962 · 10−6 1.9160 · 10−6 6.1171 · 10−8 3.7173 · 10−8

Table B.5: Influence of the coupling terms for Test Case 4 in linear transient heat equation dependingon the number of the considered time substeps

118 B. Linear transient heat equation

Number ofSbK SbM SoK SoMoverlayed elements

2 2.5958 · 10−6 1.9151 · 10−6 6.1160 · 10−8 3.6967 · 10−8

3 2.2450 · 10−6 1.7838 · 10−6 8.4728 · 10−8 5.4759 · 10−8

4 2.1265 · 10−6 1.7497 · 10−6 9.3766 · 10−8 6.2074 · 10−8

5 2.0699 · 10−6 1.7382 · 10−6 9.8084 · 10−8 6.5663 · 10−8

6 2.0378 · 10−6 1.7341 · 10−6 1.0047 · 10−7 6.7666 · 10−8

10 1.9859 · 10−6 1.7347 · 10−6 1.0397 · 10−7 7.0651 · 10−8

20 1.9573 · 10−6 1.7428 · 10−6 1.0547 · 10−7 7.1935 · 10−8

50 1.9442 · 10−6 1.7509 · 10−6 1.0589 · 10−7 7.2297 · 10−8

Table B.6: Influence of the coupling terms for Test Case 4 in linear transient heat equation dependingon the number of the overlayed elements per each base element

BellSbK SbM SoK SoMsize σ

0.02 0 2.6179 · 10−4 4.4451 · 10−28 1.4662 · 10−8

0.015 0 1.4532 · 10−4 2.5637 · 10−28 2.5293 · 10−8

0.01 0 6.2189 · 10−5 9.1848 · 10−29 5.2164 · 10−8

0.0075 0 3.3194 · 10−5 1.6865 · 10−29 8.1944 · 10−8

0.005 0 1.2873 · 10−5 1.2465 · 10−31 1.2932 · 10−7

0.002 0 7.2315 · 10−6 3.0414 · 10−37 8.9296 · 10−7

Table B.7: Influence of the reduction of the bell size for Test Case 3 in linear transient heat equation

119

Appendix C

Nonlinear stationary heat equation

Convergence results of the test cases for nonlinear stationary heat equation

Number relative L2-errorof overlayed

Method 1 Method 2 Method 3 Method 4elements

2 6.2500 · 10−6 6.2500 · 10−6 6.2500 · 10−6 6.2500 · 10−6

3 1.2346 · 10−6 1.2346 · 10−6 1.2346 · 10−6 1.2346 · 10−6

4 3.9062 · 10−7 3.9062 · 10−7 3.9063 · 10−7 3.9063 · 10−7

5 1.6000 · 10−7 1.6000 · 10−7 1.6000 · 10−7 1.6000 · 10−7

10 1.0000 · 10−8 1.0000 · 10−8 1.0000 · 10−8 1.0000 · 10−8

20 6.2500 · 10−10 6.2500 · 10−10 6.2500 · 10−10 6.2500 · 10−10

30 1.2346 · 10−10 1.2346 · 10−10 1.2346 · 10−10 1.2346 · 10−10

40 3.9063 · 10−11 3.9063 · 10−11 3.9063 · 10−11 3.9063 · 10−11

50 1.6000 · 10−11 1.6000 · 10−11 1.6000 · 10−11 1.6000 · 10−11

60 7.7160 · 10−12 7.7160 · 10−12 7.7160 · 10−12 7.7160 · 10−12

70 4.1649 · 10−12 4.1649 · 10−12 4.1649 · 10−12 4.1649 · 10−12

80 2.4414 · 10−12 2.4414 · 10−12 2.4414 · 10−12 2.4414 · 10−12

90 1.5242 · 10−12 1.5242 · 10−11 1.5242 · 10−12 1.5242 · 10−12

100 1.0000 · 10−12 1.0000 · 10−12 1.0000 · 10−12 1.0000 · 10−12

Table C.1: Convergence studies on spatial refinement for Test case 1 (full overlay) in nonlinearstationary studies

120 C. Nonlinear stationary heat equation



2 6.2299 · 10−6 6.2299 · 10−6 6.2299 · 10−6 6.2299 · 10−6

3 1.2328 · 10−6 1.2328 · 10−6 1.2328 · 10−6 1.2328 · 10−6

4 3.9031 · 10−7 3.9031 · 10−7 3.9031 · 10−7 3.9031 · 10−7

5 1.5992 · 10−7 1.5992 · 10−7 1.5992 · 10−7 1.5992 · 10−7

10 9.9987 · 10−9 9.9987 · 10−9 9.9987 · 10−9 9.9987 · 10−9

20 6.2498 · 10−10 6.2498 · 10−10 6.2498 · 10−10 6.2498 · 10−10

30 1.2346 · 10−10 1.2346 · 10−10 1.2346 · 10−10 1.2346 · 10−10

40 3.9062 · 10−11 3.9062 · 10−11 3.9062 · 10−11 3.9062 · 10−11

50 1.6000 · 10−11 1.6000 · 10−11 1.6000 · 10−11 1.6000 · 10−11

60 7.7160 · 10−12 7.7160 · 10−12 7.7160 · 10−12 7.7160 · 10−12

70 4.1649 · 10−12 4.1649 · 10−12 4.1649 · 10−12 4.1649 · 10−12

80 2.4414 · 10−12 2.4414 · 10−12 2.4414 · 10−12 2.4414 · 10−12

90 1.5242 · 10−12 1.5242 · 10−12 1.5242 · 10−12 1.5242 · 10−12

100 1.0000 · 10−12 1.0000 · 10−12 1.0000 · 10−12 1.0000 · 10−12

Table C.2: Convergence studies on spatial refinement for Test case 2 (full overlay) in nonlinearstationary studies



2 9.0315 · 10−4 9.0315 · 10−4 9.0315 · 10−4 9.0315 · 10−4

3 1.8385 · 10−4 1.8385 · 10−4 1.8385 · 10−4 1.8385 · 10−4

4 5.8811 · 10−5 5.8811 · 10−5 5.8811 · 10−5 5.8811 · 10−5

5 2.4219 · 10−5 2.4219 · 10−5 2.4219 · 10−5 2.4219 · 10−5

10 1.5355 · 10−6 1.5355 · 10−6 1.5355 · 10−6 1.5355 · 10−6

20 1.0620 · 10−7 1.0620 · 10−7 1.0620 · 10−7 1.0620 · 10−7

30 2.9322 · 10−8 2.9322 · 10−8 2.9322 · 10−8 2.9322 · 10−8

40 1.6307 · 10−8 1.6307 · 10−8 1.6307 · 10−8 1.6307 · 10−8

50 1.2724 · 10−8 1.2724 · 10−8 1.2724 · 10−8 1.2724 · 10−8

60 1.1424 · 10−8 1.1424 · 10−8 1.1424 · 10−8 1.1424 · 10−8

70 1.0861 · 10−8 1.0861 · 10−8 1.0861 · 10−8 1.0861 · 10−8

80 1.0584 · 10−8 1.0584 · 10−8 1.0584 · 10−8 1.0584 · 10−8

90 1.0435 · 10−8 1.0435 · 10−8 1.0435 · 10−8 1.0435 · 10−8

100 1.0348 · 10−8 1.0348 · 10−8 1.0348 · 10−8 1.0348 · 10−8

Table C.3: Convergence studies on spatial refinement for Test case 3 (partial overlay) in nonlinearstationary studies

121

Number relative L2-errorof base


40 1.2724 · 10−8 1.2724 · 10−8 1.2724 · 10−8 1.2724 · 10−8

50 3.6639 · 10−9 3.6639 · 10−9 3.6639 · 10−9 3.6639 · 10−9

60 1.2816 · 10−9 1.2816 · 10−9 1.2816 · 10−9 1.2816 · 10−9

70 5.3451 · 10−10 5.3451 · 10−10 5.3451 · 10−10 5.3451 · 10−10

80 2.5815 · 10−10 2.5815 · 10−10 2.5815 · 10−10 2.5815 · 10−10

90 1.4007 · 10−10 1.4007 · 10−10 1.4007 · 10−10 1.4007 · 10−10

100 8.3170 · 10−11 8.3170 · 10−11 8.3170 · 10−11 8.3170 · 10−11

Table C.4: Convergence studies on spatial refinement for Test case 3 (partial overlay) in nonlinearstationary studies, keeping the number of the overlayed elements as 50

Studies on the precision of the computation of Jacobian for nonlinear stationaryheat equation

Number of Relative Newton Gauss- Seideloverlayed elements L2-error iterations iterations (each)

2 6.2500 · 10−6 8 2

3 1.2346 · 10−6 8 2

4 3.9062 · 10−7 8 2

5 1.6000 · 10−7 8 2

10 1.0000 · 10−8 8 2

20 6.2500 · 10−10 8 2

30 1.2346 · 10−10 8 2

40 3.9062 · 10−11 8 2

50 1.6000 · 10−11 8 2

60 7.7160 · 10−12 8 2

70 4.1649 · 10−12 8 2

80 2.4414 · 10−12 8 2

90 1.5542 · 10−12 8 2

100 1.0000 · 10−12 8 2

Table C.5: Convergence data with number of necessary iterations for Test Case 1 considering allterms (Method 1)


Number of Relative Newton Gauss- Seideloverlayed elements L2-error iterations iterations (in average each)

2 6.2505 · 10−6 32 23

3 1.2344 · 10−6 25 14

4 3.9069 · 10−7 22 14

5 1.6003 · 10−7 20 14

10 1.0000 · 10−8 17 15

20 6.2541 · 10−10 16 15

30 1.2358 · 10−10 16 15

40 3.9123 · 10−11 16 14

50 1.6036 · 10−11 16 14

60 7.7404 · 10−12 16 14

70 4.1824 · 10−12 16 14

80 2.4546 · 10−12 16 14

90 1.5345 · 10−12 16 14

100 1.0084 · 10−12 16 14

Table C.6: Convergence data with number of necessary iterations for Test Case 1 neglecting the K ′bb

and K ′oo (Method 1)

Number of Relative Newton Gauss- Seideloverlayed elements L2-error iterations iterations

2 6.2500 · 10−6 1 10

3 1.2346 · 10−6 1 10

4 3.9063 · 10−7 1 10

5 1.6000 · 10−7 1 10

10 1.0000 · 10−8 1 10

20 6.2500 · 10−10 1 10

30 1.2346 · 10−10 1 10

40 3.9063 · 10−11 1 10

50 1.6000 · 10−11 1 10

60 7.7160 · 10−12 1 10

70 4.1649 · 10−12 1 10

80 2.4414 · 10−12 1 10

90 1.5242 · 10−12 1 10

100 1.0000 · 10−12 1 10


123


2 6.2500 · 10−6 1 68

3 1.2346 · 10−6 1 66

4 3.9062 · 10−7 1 59

5 1.6000 · 10−7 1 65

10 1.0000 · 10−8 1 68

20 6.2500 · 10−10 1 68

30 1.2346 · 10−10 1 68

40 3.9063 · 10−11 1 68

50 1.6000 · 10−11 1 68

60 7.7161 · 10−12 1 68

70 4.1650 · 10−12 1 68

80 2.4414 · 10−12 1 68

90 1.5242 · 10−12 1 68

100 1.0000 · 10−12 1 68

Table C.8: Convergence data with number of necessary iterations for Test Case 1 neglecting the K ′bb

and K ′oo (Method 2)

Number of Relative Gauss-Seidel Newtonoverlayed elements L2-error iterations iterations

2 6.2500 · 10−6 2 1: 8 and 3; 2:1 and 1

3 1.2346 · 10−6 2 1: 8 and 3; 2:1 and 1

4 3.9062 · 10−7 2 1: 8 and 3; 2:1 and 1

5 1.6000 · 10−7 2 1: 8 and 3; 2:1 and 1

10 1.0000 · 10−8 2 1: 8 and 3; 2:1 and 1

20 6.2500 · 10−10 2 1: 8 and 3; 2:1 and 1

30 1.2346 · 10−10 2 1: 8 and 3; 2:1 and 1

40 3.9062 · 10−11 2 1: 8 and 3; 2:1 and 1

50 1.6000 · 10−11 2 1: 8 and 3; 2:1 and 1

60 7.7160 · 10−12 2 1: 8 and 3; 2:1 and 1

70 4.1649 · 10−12 2 1: 8 and 3; 2:1 and 1

80 2.4414 · 10−12 2 1: 8 and 3; 2:1 and 1

90 1.5242 · 10−12 2 1: 8 and 3; 2:1 and 1

100 9.9996 · 10−13 2 1: 8 and 3; 2:1 and 1



Number of Relative Gauss-Seidel Newtonoverlayed elements L2-error iterations iterations

2 6.2500 · 10−6 26 1: 52 and 4; 2-26: 1 and 1

3 1.2346 · 10−6 26 1: 52 and 7; 2-26: 1 and 1

4 3.9062 · 10−7 26 1: 52 and 8; 2-26: 1 and 1

5 1.6000 · 10−7 25 1: 52 and 9; 2-25: 1 and 1

10 1.0000 · 10−8 25 1: 52 and 10; 2-25: 1 and 1

20 6.2500 · 10−10 25 1: 52 and 10; 2-25: 1 and 1

30 1.2346 · 10−10 25 1: 52 and 10; 2-25: 1 and 1

40 3.9063 · 10−11 25 1: 52 and 10; 2-25: 1 and 1

50 1.6000 · 10−11 25 1: 52 and 10; 2-25: 1 and 1

60 7.7161 · 10−12 25 1: 52 and 10; 2-25: 1 and 1

70 4.1650 · 10−12 25 1: 52 and 10; 2-25: 1 and 1

80 2.4415 · 10−12 25 1: 52 and 10; 2-25: 1 and 1

90 1.5242 · 10−12 25 1: 52 and 10; 2-25: 1 and 1

100 1.0000 · 10−12 25 1: 52 and 10; 2-25: 1 and 1

Table C.10: Convergence data with number of necessary iterations for Test Case 1 neglecting theK ′

bb and K ′oo (Method 3)

Number of Relative Mergedoverlayed elements L2-error iterations

2 6.2500 · 10−6 9

3 1.2346 · 10−6 9

4 3.9062 · 10−7 9

5 1.6000 · 10−7 9

10 1.0000 · 10−8 9

20 6.2500 · 10−10 9

30 1.2346 · 10−10 9

40 3.9062 · 10−11 9

50 1.6000 · 10−11 9

60 7.7159 · 10−12 9

70 4.1648 · 10−12 9

80 2.4413 · 10−12 9

90 1.5241 · 10−12 9

100 9.9995 · 10−13 9


125

Number of Relative Mergedoverlayed elements L2-error iterations

2 6.2500 · 10−6 65

3 1.2346 · 10−6 60

4 3.9063 · 10−7 63

5 1.6000 · 10−7 63

10 1.0000 · 10−8 64

20 6.2500 · 10−10 64

30 1.2346 · 10−10 64

40 3.9062 · 10−11 64

50 1.6000 · 10−11 64

60 7.7160 · 10−12 64

70 4.1649 · 10−12 64

80 2.4414 · 10−12 64

90 1.5241 · 10−12 64

100 9.9997 · 10−13 64

Table C.12: Convergence data with number of necessary iterations for Test Case 1 neglecting theK ′

bb and K ′oo (Method 4)

Studies on the significance of the extra terms for nonlinear stationary heatequation

GSSbK SoK SbK∆T SoK∆T SbK′∆T SoK′∆Titeration

1 0.9999 1.0000 8.4717 · 10−15 0 8.3641 · 10−15 0

2 0.9999 1.0000 8.4717 · 10−15 4.0379 · 10−12 8.3641 · 10−15 4.0378 · 10−12

Table C.13: Influence of the coupling terms for Test Case 1 in nonlinear stationary heat equationdepending on the number of performed Newton iteration (Method 1)


Number of Relative Newton Gauss- Seideloverlayed elements L2-error iterations iterations (each)

2 6.2500 · 10−6 9 2

3 1.2346 · 10−6 9 2

4 3.9062 · 10−7 9 2

5 1.6000 · 10−7 9 2

10 1.0000 · 10−8 9 2

20 6.2500 · 10−10 9 2

30 1.2346 · 10−10 9 2

40 3.9063 · 10−11 9 2

50 1.6000 · 10−11 9 2

60 7.7159 · 10−12 9 2

70 4.1648 · 10−12 9 2

80 2.4413 · 10−12 9 2

90 1.5241 · 10−12 9 2

100 9.9995 · 10−13 9 2

Table C.14: Convergence data with number of necessary iterations for Test Case 1 neglecting allextra terms (Method 1)

GSSoK SbK SoK∆T SbK∆T SoK′∆T SbK′∆Titeration

1 0 3072.9581 0 3072.9582 0 3072.9581

2 0.0052 0.0164 0 0.0155 0 0.1040

3 0.0099 0.0186 0 0.0141 0 0.1702

4 0.0130 0.0184 0 0.0071 0 0.1193

5 0.0099 0.0295 0 0.0014 0 0.0180

6 0.0164 0.3398 0 0.0002 0 0.0004

7 0.8412 1.0058 0 1.1019 · 10−7 0 1.1748 · 10−7

8 0.9999 1.0002 0 8.4736 · 10−15 0 8.3740 · 10−15

9 0.9999 1.0000 0 9.8933 · 10−29 0 9.7522 · 10−29

10 1.0000 1.0000 0 8.1188 · 10−29 0 8.1065 · 10−29

Table C.15: Influence of the coupling terms for Test Case 1 in nonlinear stationary heat equationdepending on the number of performed Gauss-Seidel iteration (Method 2)

127


2 6.2500 · 10−6 1 9

3 1.2346 · 10−6 1 9

4 3.9062 · 10−7 1 9

5 1.6000 · 10−7 1 9

10 1.0000 · 10−8 1 9

20 6.2500 · 10−10 1 9

30 1.2346 · 10−10 1 9

40 3.9062 · 10−11 1 9

50 1.6000 · 10−11 1 9

60 7.7159 · 10−12 1 9

70 4.1648 · 10−12 1 9

80 2.4413 · 10−12 1 9

90 1.5241 · 10−12 1 9

100 9.9995 · 10−13 1 9

Table C.16: Convergence data with number of necessary iterations for Test Case 1 neglecting allextra terms (Method 2)

Number ofSoK SbKGS iteration

1 0.999999978727010 0.999999978727010

2 0.999999999999967 0.999999999999967


MergedSoK SbKiteration

1 0 3072.9581

5 0.0006 0.1630

10 0.0005 0.1360

15 0.0003 0.1713

20 0.0002 0.5572

25 0.0032 0.9650

30 0.8537 0.9997

35 1.0000 1.0000

40 1.0000 1.0000


129

Appendix D

Nonlinear transient heat equation


130 D. Nonlinear transient heat equation

Basetime

step ∆tb

Overlaytime

step ∆to

Full overlay: relativeL2-error

Partial overlay:relative L2-error

Method 1

0.001 0.001 1.5186 · 10−30 2.6347 · 10−30

0.002 0.001 1.5588 · 10−30 6.6595 · 10−7

0.004 0.001 1.4814 · 10−30 5.7814 · 10−6

0.005 0.001 1.5390 · 10−30 9.8766 · 10−6

0.01 0.001 1.5255 · 10−30 3.8048 · 10−5

0.02 0.001 1.3979 · 10−30 9.4738 · 10−5

0.025 0.001 1.3128 · 10−30 1.1666 · 10−4

0.05 0.001 1.2620 · 10−30 1.8173 · 10−4

Method 2

0.001 0.001 3.7364 · 10−26 5.7873 · 10−29

0.002 0.001 9.0244 · 10−27 6.6595 · 10−7

0.004 0.001 2.2830 · 10−27 5.7814 · 10−6

0.005 0.001 2.2830 · 10−27 9.8766 · 10−6

0.01 0.001 2.2830 · 10−27 3.8048 · 10−5

0.02 0.001 2.2830 · 10−27 9.4738 · 10−5

0.025 0.001 2.2830 · 10−27 1.1666 · 10−4

0.05 0.001 2.2830 · 10−27 1.8173 · 10−4

Method 3

0.001 0.001 3.2806 · 10−26 2.1882 · 10−28

0.002 0.001 7.9504 · 10−27 6.6595 · 10−7

0.004 0.001 1.9853 · 10−27 5.7814 · 10−6

0.005 0.001 1.3984 · 10−27 9.8766 · 10−6

0.01 0.001 4.0186 · 10−28 3.8048 · 10−5

0.02 0.001 1.1147 · 10−28 9.4738 · 10−5

0.025 0.001 8.6846 · 10−29 1.1666 · 10−4

0.05 0.001 3.4440 · 10−29 1.8173 · 10−4

Table D.1: Comparison of the multiscale code with the multilevel hp− code for nonlinear transienttest cases introducing the substepping for test case 2

131

Basetime

step ∆tb

Overlaytime

step ∆to

Full overlay: relativeL2-error

Partial overlay:relative L2-error

Method 1

0.001 0.001 9.6628 · 10−30 4.1749 · 10−30

0.002 0.001 2.9815 · 10−30 1.3683 · 10−9

0.004 0.001 1.5797 · 10−30 1.0364 · 10−8

0.005 0.001 1.2945 · 10−30 1.6558 · 10−8

0.01 0.001 1.2859 · 10−30 4.8833 · 10−8

0.02 0.001 1.6260 · 10−30 1.0561 · 10−7

0.025 0.001 1.9787 · 10−30 1.3232 · 10−7

0.05 0.001 2.1201 · 10−30 2.5470 · 10−7

Method 2

0.001 0.001 2.9410 · 10−22 6.9019 · 10−22

0.002 0.001 7.6244 · 10−23 1.3683 · 10−9

0.004 0.001 2.0523 · 10−23 1.0364 · 10−8

0.005 0.001 1.3644 · 10−23 1.6558 · 10−8

0.01 0.001 4.1524 · 10−24 4.8833 · 10−8

0.02 0.001 1.5581 · 10−24 1.0561 · 10−7

0.025 0.001 1.2339 · 10−24 1.3232 · 10−7

0.05 0.001 7.9934 · 10−25 2.5470 · 10−7

Method 3

0.001 0.001 3.6848 · 10−22 7.8541 · 10−22

0.002 0.001 9.4833 · 10−23 1.3683 · 10−9

0.004 0.001 2.5203 · 10−23 1.3683 · 10−9

0.005 0.001 1.6645 · 10−23 1.6558 · 10−8

0.01 0.001 4.9103 · 10−24 4.8833 · 10−8

0.02 0.001 1.7369 · 10−24 1.0561 · 10−7

0.025 0.001 1.3813 · 10−24 1.3232 · 10−7

0.05 0.001 8.4483 · 10−25 2.5470 · 10−7

Table D.2: Comparison of the multiscale code with the multilevel hp− code for nonlinear transienttest cases introducing the substepping for test case 3


Convergence studies

Numberof over-layed

elements

Method 1 Method 2 Method 3

2 5.4441 · 10−6 5.4441 · 10−6 5.4441 · 10−6

3 1.0779 · 10−6 1.0779 · 10−6 1.0779 · 10−6

4 3.4135 · 10−7 3.4135 · 10−7 3.4135 · 10−7

5 1.3987 · 10−7 1.3987 · 10−7 1.3987 · 10−7

10 8.7464 · 10−9 8.7464 · 10−9 8.7464 · 10−9

20 5.4672 · 10−10 5.4672 · 10−10 5.4672 · 10−10

30 1.0800 · 10−10 1.0800 · 10−10 1.0800 · 10−10

40 3.4171 · 10−11 3.4171 · 10−11 3.4171 · 10−11

50 1.3997 · 10−11 1.3997 · 10−11 1.3997 · 10−11

100 8.7479 · 10−13 8.7479 · 10−13 8.7479 · 10−13

Table D.3: Convergence studies on the test case 2 for the spatial refinement

Numberof subtimesteps


2 2.0409 · 10−3 2.0409 · 10−3 2.0409 · 10−3

4 1.3530 · 10−3 1.3530 · 10−3 1.3530 · 10−3

5 1.2411 · 10−3 1.2411 · 10−3 1.2411 · 10−3

10 1.0444 · 10−3 1.0444 · 10−3 1.0444 · 10−3

20 9.5979 · 10−4 9.5979 · 10−4 9.5979 · 10−4

35 9.2637 · 10−4 9.2637 · 10−4 9.2637 · 10−4

50 9.1348 · 10−4 9.1348 · 10−4 9.1348 · 10−4

100 8.9880 · 10−4 8.9880 · 10−4 8.9880 · 10−4

Table D.4: Convergence studies on the test case 3 for time refinement

133

Numberof subtimesteps


2 9.5979 · 10−4 9.5979 · 10−4 9.5979 · 10−4

3 2.1841 · 10−4 2.1841 · 10−4 2.1841 · 10−4

4 8.1966 · 10−5 8.1966 · 10−5 8.1966 · 10−5

5 4.1363 · 10−5 4.1363 · 10−5 4.1363 · 10−5

10 9.9012 · 10−6 9.9012 · 10−6 9.9012 · 10−6

20 6.1362 · 10−6 6.1362 · 10−6 6.1362 · 10−6

30 5.6205 · 10−6 5.6205 · 10−6 5.6205 · 10−6

40 5.4534 · 10−6 5.4534 · 10−6 5.4534 · 10−6

50 5.3784 · 10−6 5.3784 · 10−6 5.3784 · 10−6

100 5.2807 · 10−6 5.2807 · 10−6 5.2807 · 10−6

Table D.5: Convergence studies on the test case 3 for space refinement

Studies on the significance of the extra terms for nonlinear transient heatequation

Number ofoverlayedelements

Realtive L2-errorAll terms

Newtoniterations


2 5.4441 · 10−6 3 8 GS (each)

3 1.0779 · 10−6 3 8 GS (each)

4 3.4135 · 10−7 3 8 GS (each)

5 1.3987 · 10−7 3 8 GS (each)

10 8.7464 · 10−9 3 8 GS (each)

20 5.4672 · 10−10 3 8 GS (each)

30 1.0800 · 10−10 3 8 GS (each)

40 3.4171 · 10−11 3 8 GS (each)

50 1.3997 · 10−11 3 8 GS (each)

100 8.7479 · 10−13 3 8 GS (each)

K ′bb = K ′oo = 0

2 5.4441 · 10−6 4 8 GS (each)

3 1.0779 · 10−6 4 8 GS (each)

4 3.4135 · 10−7 4 8 GS (each)

5 1.3987 · 10−7 4 8 GS (each)

10 8.7464 · 10−9 4 8 GS (each)

20 5.4672 · 10−10 4 8 GS (each)

30 1.0800 · 10−10 4 8 GS (each)

40 3.4171 · 10−11 4 8 GS (each)

50 1.3997 · 10−11 4 8 GS (each)

100 8.7479 · 10−13 4 8 GS (each)

M ′bb = M ′bb = M ′oo = M ′oo = 0

2 5.4441 · 10−6 4 8 GS (each)

3 1.0779 · 10−6 4 8 GS (each)


4 3.4135 · 10−7 4 8 GS (each)

5 1.3987 · 10−7 4 8 GS (each)

10 8.7464 · 10−9 4 8 GS (each)

20 5.4672 · 10−10 4 8 GS (each)

30 1.0800 · 10−10 4 8 GS (each)

40 3.4171 · 10−11 4 8 GS (each)

50 1.3997 · 10−11 4 8 GS (each)

100 8.7479 · 10−13 4 8 GS (each)

Table D.6: Convergence data with number of necessary iterations for Test Case 2 neglecting someterms in Jacobian (Method 1)



Newtoniterations


2 5.4441 · 10−6 7 2 GS (each)

3 1.0779 · 10−6 7 2 GS (each)

4 3.4133 · 10−7 7 2 GS (each)

5 1.3986 · 10−7 7 2 GS (each)

10 8.7427 · 10−9 7 2 GS (each)

20 5.4577 · 10−10 7 2 GS (each)

30 1.0757 · 10−10 7 2 GS (each)

40 3.3933 · 10−11 7 2 GS (each)

50 1.3844 · 10−11 7 2 GS (each)

100 8.3715 · 10−13 7 2 GS (each)

Table D.7: Convergence data with number of necessary iterations for Test Case 2 neglecting allcoupling terms originating from the coupling Jacobian (Method 1)



Gauss-Seidel

iterations

Newton iterations

2 5.4441 · 10−6 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

3 1.0779 · 10−6 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

4 3.4135 · 10−7 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

5 1.3987 · 10−7 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

10 8.7464 · 10−9 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

20 5.4672 · 10−10 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

30 1.0800 · 10−10 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

40 3.4171 · 10−11 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

50 1.3997 · 10−11 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

100 8.7479 · 10−13 8 1: 3 and 2; 2: 2 and 2; 3-8: 1 and 1

K ′bb = K ′oo = 0

2 5.4441 · 10−6 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

3 1.0779 · 10−6 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

135

4 3.4135 · 10−7 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

5 1.3987 · 10−7 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

10 8.7464 · 10−9 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

20 5.4672 · 10−10 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

30 1.0800 · 10−10 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

40 3.4171 · 10−11 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

50 1.3997 · 10−11 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

100 8.7479 · 10−13 8 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

M ′bb = M ′bb = M ′oo = M ′oo = 0

2 5.4441 · 10−6 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

3 1.0779 · 10−6 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

4 3.4135 · 10−7 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

5 1.3987 · 10−7 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

10 8.7464 · 10−9 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

20 5.4672 · 10−10 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

30 1.0800 · 10−10 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

40 3.4171 · 10−11 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

50 1.3997 · 10−11 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1

100 8.7479 · 10−13 9 1: 4 and 3; 2: 3 and 2; 3-8: 1 and 1


Number ofSoK SbK SoM SbMGS iteration

1 1.0000 0.0269 0.0000 0.9894

2 0.0584 0.0276 0.9956 0.9891

3 0.0578 0.0276 0.9955 0.9891

4 0.0578 0.0276 0.9955 0.9891

5 0.0578 0.0276 0.9955 0.9891

6 0.0578 0.0276 0.9955 0.9891

7 0.0578 0.0276 0.9955 0.9891

Table D.9: Influence of the coupling terms for Test Case 3 in nonlinear transient heat equationdepending on the number of performed Gauss-Seidel iteration (Method 2)


Number of Relative All K ′bb = 0 M ′bb = 0; M ′bb = 0

overlayed elements L2-error terms K ′oo = 0 M ′oo = 0; M ′oo = 0

2 5.4441 · 10−6 7 8 9

3 1.0779 · 10−6 7 8 9

4 3.4135 · 10−7 7 8 9

5 1.3987 · 10−7 7 8 9

10 8.7464 · 10−9 7 8 9

20 5.4672 · 10−10 7 8 9

30 1.0800 · 10−10 7 8 9

40 3.4171 · 10−11 7 8 9

50 1.3997 · 10−11 7 8 9

100 8.7479 · 10−13 7 8 9


Number ofSoK SbK SoM SbMGS iteration

1 0.0000 0.0007 0.0000 0.0245

2 1.0691 0.0267 18.2242 0.9556

3 0.0583 0.0276 1.0037 0.9886

4 0.0578 0.0276 0.9956 0.9891

5 0.0578 0.0276 0.9955 0.9891

6 0.0578 0.0276 0.9955 0.9891

7 0.0578 0.0276 0.9955 0.9891

Table D.11: Influence of the coupling terms for Test Case 3 in nonlinear transient heat equationdepending on the number of performed Gauss-Seidel iteration (Method 3)

137

Appendix E

Compact disc

In the attached CD the following content can be found:

• multiscale hp− d MatLAB codes for one-dimensional heat problems

• this docmentation, written in LATEX together with the respective figures.

138 E. Compact disc

LIST OF FIGURES 139

List of Figures

2.1 Type of the boundary conditions (BCs) . . . . . . . . . . . . . . . . . . . . . 6

2.2 Hierarchic basis functions in one dimension . . . . . . . . . . . . . . . . . . . 11

3.1 Different h−refinement techniques (adopted from [Zienkiewicz, 2005]) . . . . 18

3.2 hp− d concept for one-dimensional problems . . . . . . . . . . . . . . . . . . 20

3.3 Multiscale hp− d meshes construction (adopted from [Zander et al., 2015]) . 21

3.4 Multilevel hp meshes construction adopted from [Zander et al., 2015] . . . . . 22

4.1 Integration on the constructed base and overlay domains . . . . . . . . . . . . 26

4.2 Temperature distribution for the constant material distribution and sinusoidalload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Total temperature distribution for the constant material distribution and si-nusoidal load compared to analytical solution . . . . . . . . . . . . . . . . . . 29

4.4 Temperature distribution for the linear material distribution and constant load 30

4.5 Total temperature distribution for the linear material distribution and constantload compared to analytical solution . . . . . . . . . . . . . . . . . . . . . . . 31

4.6 Error in the energy norm for the linear material distribution and constant load 31

4.7 Temperature distribution for the sinusoidal material distribution and sinu-soidal load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.8 Total temperature distribution for the sinusoidal material distribution andsinusoidal load compared to analytical solution . . . . . . . . . . . . . . . . . 33

4.9 Error in the energy norm for sinusoidal material distribution and sinusoidal load 33

4.10 Influence of the neglecting of the coupling terms on the convergence behaviorfor test case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.11 Time sub stepping within multiscale hp− d approach . . . . . . . . . . . . . . 38

4.12 Development of hp− d solution within one base time step using Method 1 . . 40

140 LIST OF FIGURES

4.13 Development of hp− d solution within one base time step using Method 2 . . 40

4.14 Computation of the initial condition for the base and overlay meshes . . . . . 42

4.15 Workflow for sub stepping within one base time step. . . . . . . . . . . . . . . 43

4.16 Error introduced by the increase of the base time step ∆tb for a full overlay . 46

4.17 Error introduced by the increase of the base time step ∆tb for a partial overlay 46

4.18 Error introduced by the increase of the base time step ∆tb for a moving overlay 47

4.19 Convergence studies on time refinement . . . . . . . . . . . . . . . . . . . . . 48

4.20 Convergence studies on space refinement . . . . . . . . . . . . . . . . . . . . . 48

4.21 Influence of the width of the Gaussian bell on the solution field . . . . . . . . 50

4.22 Convergence studies on the spatial refinement for Test case 1 . . . . . . . . . 70



4.25 Convergence studies on the spatial refinement for Test case 3, keeping thenumber of the overlayed elements as 50 . . . . . . . . . . . . . . . . . . . . . 72

4.26 Number of calls to “\” against the convergence criteria for Test case 1 withfull overlay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73




4.30 Convergence studies on time refinement for Test case 3 considering 40 baseelements, each overlayed with 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.31 Convergence studies on space refinement for Test case 3 considering 10 basetime steps and 20 overlay sub steps . . . . . . . . . . . . . . . . . . . . . . . 103

4.32 Number of the calls of “\” against the convergence criteria for Test case 2 withfull overlay, recorded at the last time step . . . . . . . . . . . . . . . . . . . . 104

4.33 Number of the calls of “\” against the convergence criteria for Test case 3 withfull overlay, recorded at the last time step . . . . . . . . . . . . . . . . . . . . 104

LIST OF TABLES 141

List of Tables

2.1 Methods belonging to the generalized trapezoidal family . . . . . . . . . . . . 13

4.1 Verification of the multiscale code with the multilevel hp-code . . . . . . . . . 28

4.3 Influence of the neglection of the coupling terms for Test Case 3 . . . . . . . 35

4.5 Verification of the multiscale code with the multilevel hp-code for transienttest cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Error in the energy norm depending on the increase of the base time step ∆tbfor test cases 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.8 Multiscale hp−d finite element matrices for nonlinear stationary heat equationapproximated by Newton-Gauss-Seidel procedure . . . . . . . . . . . . . . . . 57

4.10 Multiscale hp− d finite element matrices for nonlinear heat equation approxi-mated by Gauss-Seidel-Newton procedure . . . . . . . . . . . . . . . . . . . . 61

4.12 Multiscale hp−s finite element matrices for a nonlinear heat equation approx-imated by a one-step Gauss-Seidel-Newton procedure . . . . . . . . . . . . . . 62

4.13 Verification of the multiscale hp − d code with the multilevel hp−code fornonlinear stationary test cases considering a full overlay . . . . . . . . . . . . 68

4.14 Verification of the multiscale hp− d code with the multilevel hp-code for non-linear stationary test cases considering a partial overlay . . . . . . . . . . . . 69

4.15 Finite element matrices for nonlinear transient heat equation approximated byNewton-Gauss-Seidel procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.16 Finite element matrices for nonlinear transient heat equation approximated byGauss-Seidel-Newton procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.17 Finite element matrices for nonlinear transient heat equation approximated byone-step Gauss-Seidel-Newton procedure . . . . . . . . . . . . . . . . . . . . . 95

4.18 Verification of the multiscale code with the multilevel hp-code for nonlineartransient test cases considering full overlay . . . . . . . . . . . . . . . . . . . . 98

4.19 Verification of the multiscale code with the multilevel hp-code for nonlineartransient test cases considering partial overlay . . . . . . . . . . . . . . . . . . 99

142 LIST OF TABLES

4.20 Verification of the multiscale code with the multilevel hp-code for nonlineartransient test cases considering moving overlay . . . . . . . . . . . . . . . . . 99

A.1 Influence of the coupling terms for Test Case 2 in linear stationary heat equa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

A.2 Influence of the coupling terms for Test Case 2 on the convergence behaviourin linear stationary heat equation . . . . . . . . . . . . . . . . . . . . . . . . 112

A.3 Influence of the coupling terms for Test Case 3 in linear stationary heat equa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.4 Influence of the coupling terms for Test Case 3 on the convergence behaviourin linear stationary heat equation . . . . . . . . . . . . . . . . . . . . . . . . 113

B.1 Convergence studies on time refinement for Test case 3 . . . . . . . . . . . . 115

B.2 Convergence studies on spatial refinement for Test case 3 . . . . . . . . . . . 116

B.3 Influence of the coupling terms for Test Case 3 in linear transient heat equationdepending on the number of the considered time substeps . . . . . . . . . . . 116

B.4 Influence of the coupling terms for Test Case 3 in linear transient heat equationdepending on the number of the overlayed elements per each base element . . 117

B.5 Influence of the coupling terms for Test Case 4 in linear transient heat equationdepending on the number of the considered time substeps . . . . . . . . . . . 117

B.6 Influence of the coupling terms for Test Case 4 in linear transient heat equationdepending on the number of the overlayed elements per each base element . . 118

B.7 Influence of the reduction of the bell size for Test Case 3 in linear transientheat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

C.1 Convergence studies on spatial refinement for Test case 1 (full overlay) innonlinear stationary studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

C.2 Convergence studies on spatial refinement for Test case 2 (full overlay) innonlinear stationary studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.3 Convergence studies on spatial refinement for Test case 3 (partial overlay) innonlinear stationary studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.4 Convergence studies on spatial refinement for Test case 3 (partial overlay) innonlinear stationary studies, keeping the number of the overlayed elements as50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C.5 Convergence data with number of necessary iterations for Test Case 1 consid-ering all terms (Method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C.6 Convergence data with number of necessary iterations for Test Case 1 neglect-ing the K ′bb and K ′oo (Method 1) . . . . . . . . . . . . . . . . . . . . . . . . . 122

LIST OF TABLES 143







C.13 Influence of the coupling terms for Test Case 1 in nonlinear stationary heatequation depending on the number of performed Newton iteration (Method 1) 125

C.14 Convergence data with number of necessary iterations for Test Case 1 neglect-ing all extra terms (Method 1) . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.15 Influence of the coupling terms for Test Case 1 in nonlinear stationary heatequation depending on the number of performed Gauss-Seidel iteration (Method2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C.16 Convergence data with number of necessary iterations for Test Case 1 neglect-ing all extra terms (Method 2) . . . . . . . . . . . . . . . . . . . . . . . . . . 127



D.1 Comparison of the multiscale code with the multilevel hp− code for nonlineartransient test cases introducing the substepping for test case 2 . . . . . . . . 130

D.2 Comparison of the multiscale code with the multilevel hp− code for nonlineartransient test cases introducing the substepping for test case 3 . . . . . . . . 131

D.3 Convergence studies on the test case 2 for the spatial refinement . . . . . . . 132

D.4 Convergence studies on the test case 3 for time refinement . . . . . . . . . . . 132

D.5 Convergence studies on the test case 3 for space refinement . . . . . . . . . . 133

D.6 Convergence data with number of necessary iterations for Test Case 2 neglect-ing some terms in Jacobian (Method 1) . . . . . . . . . . . . . . . . . . . . . 134

144 LIST OF TABLES

D.7 Convergence data with number of necessary iterations for Test Case 2 neglect-ing all coupling terms originating from the coupling Jacobian (Method 1) . . 134


D.9 Influence of the coupling terms for Test Case 3 in nonlinear transient heat equa-tion depending on the number of performed Gauss-Seidel iteration (Method2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135


D.11 Influence of the coupling terms for Test Case 3 in nonlinear transient heat equa-tion depending on the number of performed Gauss-Seidel iteration (Method3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

BIBLIOGRAPHY 145

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