anisotropic h-adaptive (ah-adaptive) finite element …
TRANSCRIPT
The Pennsylvania State University
The Graduate School
Department of Mechanical Engineering
ANISOTROPIC H-ADAPTIVE (AH-ADAPTIVE) FINITE
ELEMENT SCHEME FOR THREE-DIMENSIONAL
MULTI-SCALE ANALYSES
A Thesis in
Mechanical Engineering
by
Shih-Horng Tsau
c© 2006 Shih-Horng Tsau
Submitted in Partial Fulfillmentof the Requirements
for the Degree of
Doctor of Philosophy
December 2006
The thesis of Shih-Horng Tsau has been reviewed and approved* by the following: Panagiotis Michaleris Associate Professor of Mechanical Engineering Thesis Adviser Chair of Committee Ashok D. Belegundu Professor of Mechanical Engineering Eric Mockensturm Associate Professor of Mechanical Engineering Francesco Costanzo Associate Professor of Engineering Mechanics Karen A. Thole Professor of Mechanical Engineering Head of the Department of Mechanical Engineering *Signatures are on file in the Graduate School.
iii
Abstract
Using a static mesh in a multi-scale simulation, such as welding, requires many
fine elements from the start of the analysis. The mesh needs to be fine throughout the
entire simulation in both transverse and longitudinal directions to capture high gradients.
Isotropic adaptive meshing performs simultaneous coarsening, and refining, in all spatial
dimensions. Application of isotropic adaptive meshing allows the use of a coarse mesh
as the analysis begins and it refines as needed in all directions during the simulation.
However, because of the nature of isotropic refinement, elements need to remain fine in
all dimensions even if the gradient is high in only one direction. In this work, an effi-
cient Anisotropic h-Adaptive (abbreviated AH-adaptive) FEA method is developed that
performs independent refining and coarsening among all spatial dimensions. Application
of the anisotropic h-adaptive meshing allows the use of a coarse mesh as the analysis
starts. If there is one direction in which the gradient is much smoother than the others,
the mesh coarsens in the corresponding direction, thus reducing the number of DOFs by
n12 in 2D analyses, and n
13 in 3D analyses.
Dependent (also referred to as “constraint”) nodes occur when h-adaptive refine-
ment strategy is applied. The DOFs (degrees of freedom) on these dependent nodes must
be separated from the original system of algebraic equations. Only the unconstrained
(also referred to as “free”) DOFs can exist in the real equation system to solve and there-
fore yield accurate solution fileds. To deal with the dependent DOFs, several methods
can be applied for the numerical computations. A comparison between Condensation
iv
and Recovery Method, Lagrange Multiplier, and Penalty Method is performed. And the
Condensation and Recovery Method is chosen to be applied in the AH-adaptive FEA
scheme to maximize the computational efficiency.
Highlights from this research include important contributions such as: 1) simpli-
fied gradient calculations for each element, 2) nonzero fill-in effects induced by condensing
the original algebraic equation systems, 3) moving forced refinements of anticipated high
gradients, 4) procedures which assist meshes with neatly coarsening elements to the al-
lowed maximum, and 5) comparisons among possible approaches for the original system
equations from a mesh which has constrained nodes.
v
Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Welding Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Adaptive Mesh Analyses . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Anisotropic h-Adaptivity . . . . . . . . . . . . . . . . . . . . 5
1.3 Motivation for Anisotropic h-Adaptivity . . . . . . . . . . . . . . . . 6
1.4 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter 2. The AH-Adaptive Thermal Analysis Scheme . . . . . . . . . . . . . . 10
2.1 Governing Equations for Transient Heat Conduction Analysis . . . . 10
2.2 Initialization of Information Arrays . . . . . . . . . . . . . . . . . . . 12
2.3 Control Criteria on Generating Elements for Self-Adapting Dynamic
Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Gradient Measure Definition . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Review of isotropic norm definitions . . . . . . . . . . . . . . 17
vi
2.4.2 Gradient measures of AH-adaptive analysis scheme . . . . . . 17
2.4.3 Evaluation of refinement level . . . . . . . . . . . . . . . . . . 19
2.5 Moving Forced Refinement . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Determination of whether/how the sphere(s) intersect elements 21
2.5.1.1 Global and Local (Iso-parametric) Coordinates of
the Sphere Center . . . . . . . . . . . . . . . . . . . 23
2.5.1.2 Identification of Whether an Element Intersects with
the Moving Spheres . . . . . . . . . . . . . . . . . . 24
2.6 Element Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.1 Mutually Coarsenable Elements . . . . . . . . . . . . . . . . . 28
2.6.2 Transferring Data . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.3 Processing the Nodes . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.4 Processing the Elements . . . . . . . . . . . . . . . . . . . . . 33
2.6.5 Sequence of Element Coarsening . . . . . . . . . . . . . . . . 34
2.7 Element Refining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.1 Creating Entities . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.2 Transferring Solution Field and Boundary Conditions . . . . 38
2.7.3 Sequence of Element Refining . . . . . . . . . . . . . . . . . . 39
2.8 Identification of Dependent Nodes . . . . . . . . . . . . . . . . . . . 40
2.8.1 Condensation and Recovery Theory . . . . . . . . . . . . . . 40
2.9 Pre-processing for the Condensed System . . . . . . . . . . . . . . . 47
2.9.1 Determination of Constraint Equations . . . . . . . . . . . . . 47
vii
2.9.2 Nonzero Fill-ins in Sparse Tangent Matrix for AH-Adaptive
Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.9.3 Residual Array . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.10 Recovering the Condensed DOFs . . . . . . . . . . . . . . . . . . . . 50
Chapter 3. Thermal Analyses Numerical Examples . . . . . . . . . . . . . . . . 54
3.1 Heat input model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Linear Weld Path — Comparison of Static and AH-Adaptive Analysis 54
3.2.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.3 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Combined Weld Path (Curved and Linear) — Evaluation of AH-
Adaptive Analysis Scheme . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.2 Comparison to static mesh analysis . . . . . . . . . . . . . . . 68
3.3.3 CPU scaling with model size . . . . . . . . . . . . . . . . . . 79
Chapter 4. The AH-Adaptive Mechanical Analysis Scheme . . . . . . . . . . . . 81
4.1 Governing Equations for Quasi-Static Structural Analysis . . . . . . 81
4.1.1 Small Deformation . . . . . . . . . . . . . . . . . . . . . . . . 83
4.1.2 Large Deformation . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 Initialization of Information Arrays . . . . . . . . . . . . . . . . . . . 85
viii
4.3 Control Criteria on Generating Elements for Self-Adapting Dynamic
Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Gradient Measure Definition . . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Review of isotropic norm definitions . . . . . . . . . . . . . . 88
4.4.2 Gradient measures of AH-adaptive mechanical analysis . . . . 89
4.4.3 Evaluation of refinement level . . . . . . . . . . . . . . . . . . 92
4.5 Moving Forced Refinement . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6 Element Coarsening . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.6.1 Mutually Coarsenable Elements . . . . . . . . . . . . . . . . . 94
4.6.2 Transferring Data . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6.3 Processing the Nodes and Elements . . . . . . . . . . . . . . . 95
4.6.4 Sequence of Element Coarsening . . . . . . . . . . . . . . . . 96
4.7 Element Refining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.7.1 Creating Entities . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.7.2 Index and Generation Numbers . . . . . . . . . . . . . . . . . 98
4.7.3 Transferring Solution Field and Boundary Conditions . . . . 98
4.7.4 Sequence of Element Refining . . . . . . . . . . . . . . . . . . 100
4.8 Identification of Dependent Nodes . . . . . . . . . . . . . . . . . . . 100
4.9 Pre-processing for the Condensed System . . . . . . . . . . . . . . . 100
4.9.1 Determination of Constraint Equations . . . . . . . . . . . . . 100
4.9.2 Nonzero Fill-ins in Sparse Tangent Matrix for AH-Adaptive
Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.9.3 Residual Array . . . . . . . . . . . . . . . . . . . . . . . . . . 102
ix
4.10 Gauss Point Quantities Balancing . . . . . . . . . . . . . . . . . . . . 102
4.11 Recovering the Condensed DOFs . . . . . . . . . . . . . . . . . . . . 107
4.12 Convergence Efficiency Improving Strategy . . . . . . . . . . . . . . 109
Chapter 5. Mechanical Analyses Numerical Examples . . . . . . . . . . . . . . . 111
5.1 Linear Weld Path — Comparison of Static and AH-Adaptive Analysis 111
5.1.1 Hardware and Software . . . . . . . . . . . . . . . . . . . . . 112
5.1.2 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Combined Weld Path (Curved and Linear) — Evaluation of AH-
Adaptive Analysis Scheme . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . 118
Chapter 6. Comparisons between using Condensation Theory, Lagrange Multiplier
and Penalty Method for Constrained DOFs . . . . . . . . . . . . . . 127
6.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 Determination of Constraint Equations . . . . . . . . . . . . . . . . . 128
6.3 Condensation and Recovery Theory . . . . . . . . . . . . . . . . . . . 129
6.3.1 System Condensing . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3.3 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Lagrange Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.4.1 Equation Derivation . . . . . . . . . . . . . . . . . . . . . . . 135
6.4.2 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
x
6.5 Penalty Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.5.1 Utilizing Penalty Number in a System . . . . . . . . . . . . . 138
6.5.2 Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Chapter 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xi
List of Tables
3.1 Comparison between the static and the AH-adaptive analyses on the model 65
3.2 Statistics of the AH-adaptive analysis . . . . . . . . . . . . . . . . . . . 67
5.1 Comparison between the static and the AH-adaptive analyses on the model 117
5.2 Statistics of the AH-adaptive analysis . . . . . . . . . . . . . . . . . . . 126
xii
List of Figures
1.1 Concepts of element variations in isotropic and anisotropic FE schemes. 4
2.1 Flow chart of the AH-adaptive FE analysis scheme. . . . . . . . . . . . . 11
2.2 The information array of node data. . . . . . . . . . . . . . . . . . . . . 14
2.3 The information array of element data. . . . . . . . . . . . . . . . . . . . 15
2.4 Computation of gradient measures in a hex8 element. . . . . . . . . . . 20
2.5 Example of moving spheres (without combining gradient measure effect)
which guarantee the resolution at high gradient regions. . . . . . . . . . 22
2.6 Whether an element is within the moving spheres. . . . . . . . . . . . . 27
2.7 Anisotropic (and isotropic) coarsening. . . . . . . . . . . . . . . . . . . . 29
2.8 (a) Different coarsened elements (b) Index numbers associated with the
current generations (examples in r1-direction). . . . . . . . . . . . . . . 30
2.9 (a) Different new meshes using different sequences, (b) coarsening se-
quence loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Refining an element in (a) r1- (b) r2- (c) r3- direction, and the node
arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.11 Mesh containing dependent nodes. . . . . . . . . . . . . . . . . . . . . . 42
2.12 Constrained nodes between adjacent elements. . . . . . . . . . . . . . . 49
2.13 Splitting the row and column of the dependent DOF. . . . . . . . . . . . 51
2.14 Nonzero fill-in effect induced by condensing the tangent matrix. . . . . . 52
2.15 Nonzero fill-ins in the condensed matrix. . . . . . . . . . . . . . . . . . . 53
xiii
3.1 Static mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Temperature result (◦C) for the static mesh analysis. . . . . . . . . . . . 58
3.3 Cross section view for the static analysis result (◦C). . . . . . . . . . . . 59
3.4 Initial mesh for the AH-adaptive analysis. . . . . . . . . . . . . . . . . . 60
3.5 Adaptive mesh at the instance of comparison. . . . . . . . . . . . . . . . 61
3.6 Temperature result of the AH-adaptive analysis (◦C). . . . . . . . . . . 62
3.7 Zoom-in view of the AH-adaptive analysis result (◦C). . . . . . . . . . . 63
3.8 Cross section view of the AH-adaptive analysis result (◦C). . . . . . . . 64
3.9 The plate and the initial mesh. . . . . . . . . . . . . . . . . . . . . . . . 69
3.10 Temperature result (◦C) at t = 15.4 sec. . . . . . . . . . . . . . . . . . . 70
3.11 Zoom-in of the temperature result (◦C) at t = 15.4 sec. . . . . . . . . . 71
3.12 Temperature result (◦C) at t = 40.4 sec. . . . . . . . . . . . . . . . . . . 72
3.13 Zoom-in of the temperature result (◦C) at t = 40.4 sec. . . . . . . . . . 73
3.14 Temperature result (◦C) at t = 100 sec. . . . . . . . . . . . . . . . . . . 74
3.15 Zoom-in of the temperature result (◦C) at t = 100 sec. . . . . . . . . . . 75
3.16 Cross-section view from the side at t = 100 sec (◦C). . . . . . . . . . . . 76
3.17 Temperature result (◦C) at t = 3600 sec. . . . . . . . . . . . . . . . . . 77
4.1 Flow chart of the AH-adaptive FE analysis scheme. . . . . . . . . . . . . 82
4.2 The information array of node data. . . . . . . . . . . . . . . . . . . . . 86
4.3 The information array of element data. . . . . . . . . . . . . . . . . . . . 87
4.4 Gauss point interpolations for (a) coarsening, (b) refining . . . . . . . . 97
4.5 Splitting the row and column of the dependent DOF. . . . . . . . . . . . 103
xiv
4.6 Nonzero fill-in effect induced by condensing the tangent matrix. . . . . . 104
4.7 Nonzero fill-ins in the condensed matrix. . . . . . . . . . . . . . . . . . . 105
4.8 Balancing between nodal and Gauss point quantities . . . . . . . . . . . 108
5.1 Static mesh with boundary conditions . . . . . . . . . . . . . . . . . . . 113
5.2 The deformation results (mm) of the static mesh analysis. Magnification
factor = 5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.3 Initial mesh for the AH-adaptive analysis . . . . . . . . . . . . . . . . . 115
5.4 The deformation results (mm) of the adaptive mesh analysis. Magnifi-
cation factor = 5.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.5 The plate and the initial mesh. . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 Experimental buckling results. . . . . . . . . . . . . . . . . . . . . . . . 121
5.7 1st – 3rd buckling modes and pure angular distortion. . . . . . . . . . . 122
5.8 The deformation result (mm) at t = 3600 sec, with permissible gradient
(peak temperature) = 58 ◦ C. (Magnification factor = 2.5) . . . . . . . 123
5.9 The deformation result (mm) at t = 3600 sec, with permissible gradient
(peak temperature) = 400 ◦ C. (Magnification factor = 2.5) . . . . . . . 124
6.1 The nonzero components in C, A−1, CT , and the resulted matrices. . . 139
xv
Acknowledgments
I am most grateful and indebted to my thesis advisor, Dr. Panagiotis Michaleris,
for the large doses of guidance, patience, and encouragement he has shown me during
my time here at Penn State. I thank my other committee members, Dr. Ashok D.
Belegundu, Dr. Eric Mockensturm, and Dr. Francesco Costanzo, for their insightful
commentary on my work. I am also grateful and indebted to all of my labmates, for
inspiration and enlightening discussions on a wide variety of topics. I would like to
acknowledge Edward W. Reutzel (from Applied Research Lab at Pennsylvania State
University) for his suggestions on this research results.
1
Chapter 1
Introduction
1.1 Welding Simulation
1.1.1 Background
Modeling of welding distortion and residual stress has been an active research
area since the late 1970’s. Some of the first publications in weld modeling include Refs.
[1, 2, 3]. Research in the 1980’s includes the development of the “double ellipsoid” heat
input model by Goldak et al. [4], and the modeling of phase transformations [5, 6, 7].
Most of the weld modeling in the 1970’s and 1980’s utilized 2D models transverse to the
welding direction using either plane strain or generalized plane strain conditions. These
models correlated well with experimental measurements for residual stress. However,
they were not able to accurately predict angular [8] distortion or longitudinal buckling
and bowing [9]. Significant developments in weld modeling in the 1990’s included the
use of 3D moving source models [10, 11, 12], the development of sensitivity formulations
[13, 14], and the development of decoupled 2D weld process and 3D structural response
models [9].
More recent developments in the area of weld modeling include the development
of Eulerian models for simulating long, steady welds [15, 16], and the development of a
decoupled 3D weld modeling and 3D structural response approach [17]. However, Eule-
rian models are not applicable to panels with transverse stiffeners. The decoupled 3D to
2
3D approach uses reduced-length weld models to determine the plastic strain field (all
six components) resulting from welding, and then maps them to a full size 3D structural
model to determine the resulting structural distortion. The advantage of the approach
compared to previous decoupled methods is that it maps all components of plastic strain,
and therefore it accounts for angular distortion, which is dominated by the transverse
shear plastic strain component as demonstrated in [18]. Michaleris et al. [8], demon-
strated that 3D finite element models of the welding process are needed to accurately
compute angular distortion. Furthermore, 3D weld process finite element models can
easily account for the increased stiffness of plate curvature and the compliance of fixtur-
ing restraints. However, 3D finite element simulations of welding large structures require
very large models in which thermal equilibrium equations are iteratively computed for
several thousand increments [19].
1.2 Adaptive Mesh Analyses
Unlike static mesh analyses, adaptive mesh analyses dynamically generate a new
mesh whenever and wherever necessary. For large simulations such as multi-scale analy-
ses, a static scheme requires a huge computational expense because of the high number
of DOFs (degrees of freedom) on the mesh. Hence, adaptive methods provide an efficient
approach which can reduce the total CPU usage. There are three categories of the most
utilized adaptive meshing strategies [20]:
1. r-adaptivity, which only relocates the node positions, maintaining constant mesh
connectivity and node quantity,
3
2. p-adaptivity, which increases the polynomial order of the elements when a greater
degree of interpolation is necessary,
3. h-adaptivity, which generates hierarchical elements where it is necessary to acquire
more accuracy, and coarsens the elements where the solution has exceeded the
desired accuracy.
1.2.1 Comparisons
Among these options, the r-adaptivity method requires the least cost, but provides
the poorest flexibility. This is due to that the number of DOFs is fixed and the topol-
ogy (connectivity) depends on an initial mesh completely, regardless of how the nodes
are moved. Also, in terms of accuracy, it has the lowest performance because of the
same reasons. Meanwhile the p-adaptivity is still restricted by initial node coordinates
and connectivities, if it only involves changing the orders of interpolation polynomials,
without creating new elements. (To combat this, the h- and p-strategies are sometimes
combined into another hp-adaptivity category.) On the other hand, the h-adaptivity pro-
vides new entities and connectivities so that a desired accuracy can be met. However,
additional care must be taken for transferring all information onto the new mesh.
An additional limitation of the p-adaptivity is that it may induce oscillations in
the solution distribution by creating differences in the polynomial interpolation orders
of adjacent elements. Cogitating all the advantages and drawbacks leads this research
to create an efficient finite element analysis scheme using the h-adaptivity.
4
Isotropic Anisotropic (example of being refined in one specific direction)
v.s.(Must be refined to equal generations in all directions)
Generation 1Generation 1
Generation 2Generation 2
Generation 3Generation 3
r1
r2r3
Note : Colored elements correspond to the change of generation due to refinements
RR
C R
RR CC
CC
CR
R : Refining
C : Coarsening
Original (initial) element
Generation 0
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Fig. 1.1. Concepts of element variations in isotropic and anisotropic FE schemes.
5
1.2.2 Anisotropic h-Adaptivity
In this work, a three-dimensional anisotropic h-adaptive method is developed.
Figure 1.1 demonstrates the concepts on the difference between isotropic and anisotropic
refinement of elements. Isotropic h-adaptivity simultaneously refines an element in all
r1-, r2- (if 2D only) and r3-directions — that is, eight descendant elements are produced
during the refinement step in 3D analyses (or in 2D cases, four descendant elements
are produced). On the other hand, an anisotropic adaptive scheme refines elements in-
dependently in each direction in which refinement is required. In other words, for the
refinement of an element in any one direction, two elements are created. With the exam-
ples in Figure 1.1 showing refinements in one specific (r1-) direction, similar approaches
apply to refinements in other (r2-, and r3-) directions. There are a few application stud-
ies related to anisotropic FEA. Ham et al. [21] demonstrates an anisotropic Cartesian
grid method for incompressible flows. Also, Rachowicz presents an anisotropic scheme
for compressible Navier-Stokes equations [22]. Lo [23] demonstrates an anisotropic pro-
cedure for 3D tetrahedral elements.
Figure 1.1 illustrates the generations of the refined elements. Setting the initial
element(s) on the starting mesh to be of generation 0, as the refinements proceeds, the
associated number of generation for the corresponding elements are shown. Elements
(of generation N + 1) are called the descendant elements of the ancestor element (of
generation N) from which they were refined.
The anisotropic re-meshing strategy allows an element to have separate refine-
ments in any direction. This results in the flexibility of independent element generations
6
in each local direction. Thus, actually an isotropic re-meshing can be regarded as a
special case of an anisotropic scheme, when all directions are forced to have the same
generation.
The converse operation to element refinement in Figure 1.1 is element coarsening.
Thus, the isotropic approach enables the coarsening of eight elements to a single ancestor
element. The anisotropic method can coarsen two elements to one ancestor element in
any single corresponding direction.
For future references, note that after each refining or coarsening, the newly gen-
erated element(s) will be called “active element(s)”, with the old element(s) being “in-
active”.
The dynamic meshes in this paper are created using a forward adaptive meshing
scheme, i.e., in a given time increment, the analyses generate a new mesh based on
the solutions acquired in the previous time increment, and prior to solving the system
equations for the current time increment. An alternate approach is an iterative adaptive
procedure which continuously refines (or coarsens) the elements in a mesh within the
same time increment until the mesh convergence is attained. However, this approach
may be more computationally costly than the forward adaptive meshing.
1.3 Motivation for Anisotropic h-Adaptivity
The anisotropic h-adaptive method is demonstrated in the modeling of a laser
weld, where very small elements are required near the heat source, and large elements
elsewhere to model the part.
7
During welding, the temperature gradient ∇T is much higher near the heat
sources. However, upon cooling ∇T gradually goes to nearly zero, thus allowing ele-
ments to coarsen without loss of accuracy. Static mesh analyses require fine elements
along the entire heating path in all directions to capture the gradients. Adaptive meshing
allows a coarse starting mesh. Therefore, adaptive meshing is apparently more efficient
to static meshing. Also, within the period of temperature drop, an element may just need
a larger size in some direction than the other(s) because of different gradient magnitudes
among the three local directions. Thus, rather than isotropic re-meshing, anisotropic
re-meshing can provide better flexibility and efficiency for heat transfer analyses.
The advantage of anisotropic against isotropic re-meshing is even greater in me-
chanical analyses. Mechanical responses (such as plastic strain gradient) are steeper
around hear sources, but they remain high transversally and much lower longitudinally
after the heat sources pass [9]. The permanent plastic strains lead to residual stresses and
distortions. For isotropic meshing in mechanical analyses, small elements that remain
fine isotropically due to the high transverse gradients even after cooling are generated
near the heat source. Therefore, no significant coarsening can be allowed. Anisotropic
adaptive in mechanical analyses, however, can allow coarsening along the heating path
direction, and maintain enough refinement transversally. The anisotropic strategy thus
reduces DOFs by n12 in 2D analyses, and n
13 in 3D analyses compared to isotropic
adaptivity.
In welding simulations, a minimum of 3 elements/thickness and 4 elements/torch-
width in both longitudinal and transversal directions [24] is necessary around the heat
source region for sufficient numerical accuracy. Using currently available software and
8
hardware, such simulations require prohibitively costly numerical computations when
modeling applications of industrial significance. Developing an efficient computational
approach is necessary for performing large-scale moving source simulations. Besides do-
main decomposition combined with parallel computing, which requires financial invest-
ment for multiple processors to share the total CPU usage, adaptive analysis procedures
improve the efficiency for large-scale analyses.
1.4 Objective
The objective of this research is to lay the groundwork for an anisotropic h-
adaptive method which promotes the computational efficiency in simulations such as
multi-scale analyses. Some highlights of major contributions in this research include:
• Simplified gradient (or “error”, in some of the previous research of adaptive FEA)
calculations to determine how much to refine/coarsen an element.
– Easy to implement for either an orthogonal element, or even a skewed element
– Does not need to calculate the second-degree derivatives, which do not provide
meaningful information for linear elements such as Hex8 elements, of the
solution field
– Can be performed on an element independently, without more equations for
global gradients and the associated normalizations
• Revelations and investigations on nonzero fill-in effects induced by condensing the
original algebraic equation systems, due to the dependent DOFs.
9
– Information of exact nonzero positions is mandatory and important upon
using a sparse solver
• Applications of moving forced refinements along paths/at regions of anticipated
high gradients.
– consists of a direct rule to decide whether and how an element intersects with
the moving spheres
• Procedures of criteria assisting meshes with the ability to neatly coarsen elements
to the allowed maximum, even to the most coarsened mesh density as the initial
mesh if necessary.
• Comparisons among possible approaches to deal with the original system equations
from a mesh which has constrained nodes in it.
The efficiency of the AH-adaptive method is also evaluated by comparing the
results to those from static mesh analyses. And the created FEA scheme is applied on
simulations of a larger size and more DOFs.
10
Chapter 2
The AH-Adaptive Thermal Analysis Scheme
The algorithm of the AH-adaptive FE scheme is shown in the flow chart of Figure
2.1. While the adaptive scheme can be applied to any type of elements, for illustrative
purposes the following sections are based on hex8 elements wherever a specific element
type is necessary.
2.1 Governing Equations for Transient Heat Conduction Analysis
For a stationary reference frame r, at time t, the governing equation for transient
heat conduction analysis is given as follows:
ρCp∂T
∂t(r, t) = −∇r · q(r, t) + Q(r, t) in volumn V (2.1)
where ρ is the density of the flowing body, Cp is the specific heat capacity, T is the
temperature, q is the heat flux vector, Q is the internal heat generation rate, and ∇r is
the spatial gradient operator of reference frame r.
The nonlinear isotropic Fourier heat flux constitutive relation is enforced.
q = −k∇rT (2.2)
11
Read modeldata file
Read input control file
Initializationof informationarrays
inc = 1
Element coarsening
Element refining
Identification ofdependent nodes
Pre-processingfor the condensedsystem
iter = 0
Assemble theresidual andstiffness
Solve thesystem
Update thesolution vector
Recovering thecondensedDOFs
if eps(L2 norm ofincrementalsolution)< epslim
Acquire thesecondaryquantities
if time <maxtime
inc = inc + 1
Analysisfinished
No
Yes
No
Yes
iter = iter + 1
the procedures utilized in ordinary FEA (static mesh)
the procedures for AH-adaptive analysis ability
Gradientmeasurecalculations
Evaluation ofrefinementlevel
MovingForcedRefinement
control criteria for generating new elements
Fig. 2.1. Flow chart of the AH-adaptive FE analysis scheme.
12
where k is the temperature dependent thermal conductivity matrix. The initial and
boundary conditions can be found in [13].
An AH-adaptive thermal analysis involves the tasks illustrated in the following
sections.
2.2 Initialization of Information Arrays
Throughout an entire analysis, the re-meshing procedures generate new entities.
Arrays which save this information associated with nodes and elements are necessary
in order to transfer the properties between the entities and properly construct the new
mesh.
Figure 2.2 depicts the contents of the information arrays stored for each node.
They are
1. coordinates of the nodes,
2. nodal solution carried from the previous time increment,
3. node-wise boundary information.
During re-meshing, temporary arrays (of an assigned maximum size) record all
this information, while the counter-part arrays for currently active nodes are assembled
and utilized once the new mesh is determined. An “active node” is referred to as a node
assigned on any “active element” (Section 1.2.2).
Illustrated in Figure 2.3 are element information arrays, which include
1. current element generations in all directions (set to be zero for elements in an initial
mesh),
13
2. associated index numbers,
3. surface-wise and element-wise boundary condition definitions,
4. remaining need to coarsen or refine,
5. a link to the initial element from which a specific element was refined,
6. whether it is an active element.
Similarly, in addition to the temporary arrays which contain the data for the re-
meshing procedures, an array of only the active elements will remain after the re-mesh
operation is complete.
Meanwhile, due to a need to recover the properties which a refined element directly
inherits from the initial element (of the starting mesh) where it is refined from, an
additional array is generated which links to the initial element data. This serves to reduce
the computer memory usage, and the required data can be simply accessed through this
link.
With the information arrays properly defined, initial/old elements can now be re-
meshed, controlled through the following criteria which guarantee mesh densities assigned
by an analyst.
14
Node 1 Node 2 ..... Node Nmax
The temporary array for node data during re-meshing
x-coordinate
Nmax : the maximum # of total temporary nodes allowed during one loop of re-meshing
# of previous mesh nodes .....
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
Node 1 Node 2 .....
x-coordinate
# of active nodesafter re-meshing
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
The array of active node data
(With the information on what elements are active)
The nodes on active elements
* : actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
Fig. 2.2. The information array of node data.
15
Whether it is active
The initial element it comes from
..... (1:Active)0
Emax : the maximum # of total temporary elements allowed during one loop of re-meshing
Element generations
Index number associated with the generations
Elem 1 Elem 2 ..... Elem Emax# of previous mesh elements .....
Remaining generationsto refine / coarsen
*Surface-wise B.C.
*Element-wise B.C.
1 0 ..... 1
The temporary array for element data during re-meshing
Initial Elem 1 Intial Elem 2 ..... Initial Element N
Maximum generations to be refined(for all 3 local directions seperately)
The material group
Element type configuration
The base (reference) array for all initial elements
Elem 1 Elem 2 .....# of active elementsafter re-meshing
The array of active element data
*Surface-wise B.C.
*Element-wise B.C.
* : actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
b
b
b
The initial element it comes from
Element generations
Index number associated with the generations
Remaining generationsto refine / coarsen
b
b
b
a
: contains the information rows for element type, number of nodes in an elementa
: contains 3 rows, for the information corresponding to r1,r2,r3-directionsb
Fig. 2.3. The information array of element data.
16
2.3 Control Criteria on Generating Elements for Self-Adapting Dy-
namic Meshes
At the beginning of an analysis, and between two time increments, the ini-
tial/current adaptive mesh will be modified to form the new adaptive mesh. The el-
ements need to be refined, coarsened, or remain the same. The key controlling criteria
are:
1. Gradient measures, which are evaluated based on the solution field in the previous
mesh. These gradients are compared to the desired permissible gradient. Elements
are refined when the gradient measures are too high, and coarsened if too low.
2. Moving forced refinement, which are set to track the moving heat source, and force
refinement in order to guarantee sufficient element density to adequately integrate
the heat source.
2.4 Gradient Measure Definition
Except at the start of an analysis, the AH-adaptive scheme examines elements on
a current mesh after the solution field has been acquired. An element with a smoother
solution distribution which yields a smaller gradient field will tend to be given a larger
element size (by coarsening), while an element with a higher gradient demands to be
refined to a smaller size to better reflect the steep solution gradient.
17
2.4.1 Review of isotropic norm definitions
For isotropic FE analyses, in Ref [25] error norms are derived based on concepts
introduced in Refs [26, 27]. The need to isotropically refine or coarsen the elements in a
mesh is calculated according to:
‖e(i)‖ ≤ Ce(i)
hp−m+d (2.3)
where e(i) denotes the error in an element i, h is the maximum element diagonal
length, p is the order of the shape function, m is the highest order of differentiation in
the strain-displacement relation, and d equals to 1, 2, or 3 depending on the number
of dimensions. A new element size is computed after the element local error norm is
normalized by an additionally evaluated global gradient field.
While these norm equations provide a basis for isotropic refinement and coars-
ening, an anisotropic analysis requires calculations for gradients independently for all
(local r1-,r2-,r3-) directions. The gradient evaluation for the AH-adaptive FEA scheme
is developed as in the following section.
2.4.2 Gradient measures of AH-adaptive analysis scheme
The gradient measure derivation for the AH-adaptive scheme is evaluated by con-
sidering that at any interior point of an element, the solution magnitude T is calculated
by
18
T = N · T (2.4)
where N is the shape function, and T = [T1, T2, ..., T8]T (for a hex8 element) is
the nodal solution on this element. This leads to the gradient definitions of the solution
field in all directions as
∇rT =dT
dr=
d(N · T)dr
=dNdr
· T (2.5)
where the local gradient ∇r is a 3 ∗ 1 vector.
In this work, the local gradients are evaluated at the center of the isoparametric
coordinates (r1, r2, r3 = 0). Therefore, the gradient measures G for all three directions
(three scalars expressed in a vector form) are:
G = [Gr1, Gr2
, Gr3]T = |∇rT |centroid = |(dN
dr)centroid · T| (2.6)
where | | denotes the absolute values.
19
Now that the gradient measures independently in all isoparametric directions are
obtained, an element needs to be determined if it should stay at the same size, or either
be refined/coarsened.
2.4.3 Evaluation of refinement level
With the desired permissible gradient Gp set by the analyst for each specific
simulation, the need to refine or coarsen each element is determined by
Ri = log2GriGp
(2.7)
where Ri(i = 1, 2, 3) determines the need to refine or coarsen in each direction.
A positive number indicates the need to refine the element, while a negative number
implies the element can be coarsened.
As the equations show, because the measures are calculated based on local dimen-
sions, they can be easily derived for either orthogonal or skewed elements (Figure 2.4).
In addition, the computed gradient measures are completely unaffected by the difference
in global dimension sizes among all elements. Thus, this approach eliminates any need
for global normalization. Furthermore, the independent element-wise calculation also
enables the simulation to benefit from coding with multi-CPU parallelization, if desired.
20
r1, Global x
r2, Global yr3. Global z
T 1T 2
T 3 T 4
T 5T 6
T 7 T 8
Average gradientsat the element center
T 1T 2
T 3 T 4
T 5
T 6T 7
T 8
4
r1r2
r3
Fig. 2.4. Computation of gradient measures in a hex8 element.
21
2.5 Moving Forced Refinement
In the AH-adaptive analysis, a starting mesh can be set very coarse. This also
benefits an analysis by saving even more CPU usage for the time increments when the ini-
tial mesh density is already sufficient. However, for the other time increments, the Gauss
points in such a coarse mesh can be too distant from the highest energy concentration
to sufficiently integrate the heat source. Besides, a forward re-meshing technique is uti-
lized to significantly reduce the computational cost as compared to iterative re-meshing
techniques. Therefore, moving forced refinement within spherical regions moving along
with the heat source(s) is introduced to trigger and enforce sufficient element densities.
Figure 2.5 illustrates the forced refinement within dual moving spheres to guaran-
tee different degrees of refinement. The inner sphere of radius Ri controls all elements to
the finest permissible size. Meanwhile the outer sphere with radius Ro avoids an exces-
sive jump in element refinements between the inner sphere and the none-sphere region
where generally little refinement is induced. Elements in the transition region are forced
to an intermediate generation. Multiple spheres can be applied if necessary.
2.5.1 Determination of whether/how the sphere(s) intersect elements
To check if an element intersects the moving spheres, the AH-adaptive FE scheme
calculates the local coordinates of the current sphere center w.r.t. the iso-parametric
coordinate system of this element. This approach is illustrated as follows:
22
Heat Source Moving Direction
: Inner Sphere (for the most refinement)
: Outer Sphere (for medium refinement)
Instant CenterInstant Center
Fig. 2.5. Example of moving spheres (without combining gradient measure effect) whichguarantee the resolution at high gradient regions.
23
2.5.1.1 Global and Local (Iso-parametric) Coordinates of the Sphere Center
Let (x, y, z) be the known global coordinates of the sphere center in the Cartesian
system. (xi, yi, zi) with i = 1–8 are the global coordinates of the eight corner nodes on
an element. Then, first for x:
x = 18 [(1 + r1)(1 + r2)(1 + r3)x1 + (1 − r1)(1 + r2)(1 + r3)x2 + (1 − r1)(1 − r2)(1 + r3)x3
+(1 + r1)(1 − r2)(1 + r3)x4 + (1 + r1)(1 + r2)(1 − r3)x5 + (1 − r1)(1 + r2)(1 − r3)x6
+(1 − r1)(1 − r2)(1 − r3)x7 + (1 + r1)(1 − r2)(1 − r3)x8]
(2.8)
Equation (2.8) can be rearranged into the form:
x = 18 [(x1 − x2 + x3 − x4 − x5 + x6 − x7 + x8) ∗ r1 ∗ r2 ∗ r3
+(x1 − x2 + x3 − x4 + x5 − x6 + x7 − x8) ∗ r1 ∗ r2
+(x1 + x2 − x3 − x4 − x5 − x6 + x7 + x8) ∗ r2 ∗ r3
+(x1 − x2 − x3 + x4 − x5 + x6 + x7 − x8) ∗ r1 ∗ r3
+(x1 − x2 − x3 + x4 + x5 − x6 − x7 + x8) ∗ r1
+(x1 + x2 − x3 − x4 + x5 + x6 − x7 − x8) ∗ r2
+(x1 + x2 + x3 + x4 − x5 − x6 − x7 − x8) ∗ r3
+(x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8)]
(2.9)
where r1, r2, r3 are the unknown iso-parametric coordinates of the sphere center.
Similar equations can be derived for y and z separately. Thus, a system of equations is
acquired:
24
x = A1∗r1∗r2∗r3+A2∗r1∗r2+A3∗r2∗r3+A4∗r1∗r3+A5∗r1+A6∗r2+A7∗r3+A8
(2.10)
y = B1∗r1∗r2∗r3+B2∗r1∗r2+B3∗r2∗r3+B4∗r1∗r3+B5∗r1+B6∗r2+B7∗r3+B8
(2.11)
z = C1∗r1∗r2∗r3+C2∗r1∗r2+C3∗r2∗r3+C4∗r1∗r3+C5∗r1+C6∗r2+C7∗r3+C8 (2.12)
where the A’s, B’s and C’s are coefficients consisting of the xi, yi, zi(i = 1–8).
Thus, the local (iso-parametric) coordinates r1, r2, r3 can be solved numerically using
the Newton-Raphson method.
2.5.1.2 Identification of Whether an Element Intersects with the Moving
Spheres
1. If all |r1|, |r2|, |r3| ≤ 1: the sphere center is inside the element. Hence the element
must be either partially or entirely within the inner sphere (Figure 2.6(a)). There
is no need to apply the Jacobian transformation to calculate the actual distance,
and the element will be refined to the assigned maximum order.
2. Not all of |r1|, |r2|, |r3| ≤ 1: the instant center is outside the element. The actual
distance, d = |d|, between the instant center and the element needs to be evaluated
to compare with the radii of the inner and outer spheres. Each situation will fall
into one of the following three categories, which are illustrated in Figure 2.6(b1–b3)
with examples.
25
• if all |r1|, |r2|, |r3| > 1: the distance is from the instant center to the nearest
element corner node. In the specific example in Figure 2.6,
d = J · [r1 − 1, r2 − 1, r3 − 1]T (2.13)
• if only two of |r1|, |r2|, |r3| are greater than 1: the distance is between the
instant center and the nearest element edge. For the corresponding example
in the figure,
d = J · [r1 − 1, r2 − 1, 0]T (2.14)
• if only one of |r1|, |r2|, |r3| is greater than 1: the distance is between the instant
center and the projection onto the nearest element surface. For Figure 2.6(b3)
example,
d = J · [r1 − 1, 0, 0]T (2.15)
26
With the actual distance d = |d|, the AH-adaptive FE scheme compares the radii
to check whether the element intersects the moving spheres, and to what generation
the element needs forcibly refined if it intersects the sphere(s).
The preceding illustrations can furthermore be summarized as following rule: with
the values of |r1|, |r2|, |r3|, to sufficiently determine the distance vector d, the Jacobian
matrix is applied for the transformation:
d = J · [A1, A2, A3]T (2.16)
where J is the Jacobian matrix, and
An =
0,
rn − 1,
rn + 1,
if
|rn| ≤ 1
rn > 1 n = 1, 2, 3
rn < −1
(2.17)
The equation will automatically convert the iso-parametric coordinates to the
distance from the instant sphere center to the element, without needing to specifically
determine which node/edge/surface on the element to use for calculations.
The procedures illustrated in Sections 2.3 – 2.5 on control criteria provide how
many generations the elements are suggested to be refined or coarsened. The AH-
adaptive analysis will be generating new elements afterwards.
27
: Inner Sphere (for the most refinement)
: Outer Sphere (for medium refinement)
: Instant Center
r1
r2r3
All |r1|, |r2|, |r3| > 1 |r1|, |r2| > 1
|r3| <= 1|r1| > 1
|r2|, |r3| <= 1
All |r1|, |r2|, |r3| <= 1
X
X
X
(1,1,1)
(1,1,r3)(1,r2,r3)
d
d
d
(b1) (b2) (b3)
(a)
Fig. 2.6. Whether an element is within the moving spheres.
28
2.6 Element Coarsening
The AH-adaptive FE scheme processes element coarsening before refining. This
sequence reduces the number of temporary nodes and elements than the opposite way. An
element is only allowed to coarsen with its pair element if the element has the gradient
measure suggesting a larger size, and the suggested coarsening can not go below the
necessary generation enforced by the moving spheres.
2.6.1 Mutually Coarsenable Elements
Different ways of coarsening two elements in an anisotropic strategy is illustrated
in Figure 2.7. The procedure can be applied to two elements in any of the directions. In
the AH-adaptive scheme, coarsening does not depend on the sequence of how elements
were refined. Therefore, analyses are provided with more coarsening flexibility and do not
need to record the history of refinement for any element. However, totally free coarsening
could induce an undesirable situation, such as that illustrated in Figure 2.8(a1). While
the elements A, B, C are of exactly the same generations in all directions, element A
can not be coarsened with element B or element C no matter how much coarser it is
desired. Though in the example elements B, C can still coarsen mutually, this clearly
will not help coarsening element A. Note however, that the elements could be coarsened
into a single element whenever they need to, if they are coarsened in the proper manner
(Figure 2.8(a2)).
Three index numbers (I1, I2, I3), corresponding to the three local directions, are
assigned to every element and associated with current element generations (G1, G2, G3).
29
r1
r2r3
Node 1Node 2
Node 5
Node 7 Node 8
Node 3 Node 4
Node 6
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Node 1Node 2
Node 3
Node 5
Node 8Node 7
Node 6
Node 4Node 1
Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Node 1Node 2
Node 3
Node 5
Node 7 Node 8
Node 6
Node 4 Node 1Node 2dNodNodode 2de 2oddN dN d 22dd
Node 3 Node 4Node 5
NodeNooode 6ode 6odeodeoode
Node 7 Node 8
r1-coarsening
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
r2-coarsening
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
r3-coarsening
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
: Nodes at midpoints on the coarsened element May still be used by adjacent elements
Isotropic
coarsening( (Fig. 2.7. Anisotropic (and isotropic) coarsening.
30
r1
r2r3
Generation : 0(An element in the initial mesh) Generation : 1 Generation : 2
(a1)
Element A
Element B Element C
(a2)
Index : 1 Index : 1Index : 2 1234
..........
(b)
Fig. 2.8. (a) Different coarsened elements (b) Index numbers associated with the currentgenerations (examples in r1-direction).
31
Taking different generations for r1-refinement as an example, the relationship between an
index number and the corresponding generation is shown in Figure 2.8(b). The purpose
of these indices is to enable the proper coarsening of elements with other coarsenable pair
elements. The indices together with the generation numbers therefore serve as control
factors to prevent situations like Figure 2.8(a2) from happening.
Two elements A and B with
element A: Generations(GA1, GA2, GA3) and indices(IA1, IA2, IA3).
element B: Generations(GB1, GB2, GB3) and indices(IB1, IB2, IB3).
can only be mutually coarsened in direction rn if the following conditions are all
satisfied:
1. they both come from the same initial element, and
2.
GAk = GBk for k = 1, 2, 3 (2.18)
IAk = IBk for k = 1, 2, 3 and k �= n (2.19)
IAn = 2m − 1
IBn = 2m
or
IAn = 2m
IBn = 2m − 1(2.20)
where
32
GA’s: Generations of element A
GB ’s: Generations of element B
IA’s: Index numbers of element A
IB ’s: Index numbers of element B
m: any positive integer
So that the new coarsened element has the following properties
Ik = IAk(= IBk)
Gk = GAk(= GBk)for k = 1, 2, 3, k �= n (2.21)
In = m (2.22)
Gn = Gan − 1(= Gbn − 1) (2.23)
2.6.2 Transferring Data
After coarsening is finished, the original two elements will be deactivated. There-
fore, the data of the old elements need to be transferred. In an AH-adaptive thermal
analysis, there are primarily three categories of data to transfer for the finite element
entities:
1. Node-wise quantities: nodal solutions from the previous time increment, and bound-
ary conditions such as prescribed temperatures.
2. Surface-wise quantities: boundary conditions such as element convection and flux
properties.
33
3. Element-wise quantities: boundary conditions (such as body heat input), and re-
maining generations to refine/coarsen, etc.
2.6.3 Processing the Nodes
Element coarsening will never create new nodes. Neither does it involve immediate
transfer of nodal quantities. Also, a node can not be deactivated simply because of
element coarsening, as an adjacent element may still be using the node. A node can
be eliminated only when the re-meshing procedure is complete, and not any element is
using it at all.
2.6.4 Processing the Elements
The AH-adaptive scheme generates a new element after coarsening and deactivates
the original two elements. Therefore, the surface- and element-wise quantities (Section
2.6.2) need to transferred.
a) Surface-wise quantities: Upon coarsening, no new element surface will be gen-
erated as each surface will have its counterpart in one of the two original elements. The
properties on the surfaces of a new element will inherit from those of the corresponding
surfaces.
b) Element-wise quantities:
1. Boundary condition: for any direction of element coarsening, if the original two ele-
ment possess element-wise boundary conditions of the same property, the quantity
for the new element will be defined as well.
34
2. Remaining generations to refine/coarsen:
if two elements A and B are to be coarsened in rn direction, and
element A: (RA1, RA2, RA3)
element B: (RB1, RB2, RB3), where RA′s and RB
′s are the remaining genera-
tions to re-mesh in all three directions for elements A and B, respectively, the
corresponding numbers for the new element C are
RCi =
Int(RAi+RBi2 ) if i = 1 − 3 and i �= n
Int(RAi+RBi2 ) + 1 if i = n
(2.24)
where RC′s are how many generations element C needs to be re-meshed, i corre-
sponds to local directions, and Int() is taking the integer value.
2.6.5 Sequence of Element Coarsening
The AH-adaptive scheme processes all coarsenable elements in one specific local
direction (e.g. r1-direction) first. After this certain direction is finished, a second local
direction and then the third direction are operated sequentially. Due to the fact that
coarsening can only occur at two mutually coarsenable elements that are both suggested
to be coarsened, even with same results from the control criteria in Section 2.3 of how
many generations an old element should be refined/coarsened, different sequences of
coarsening may induce different elements in the new mesh. A simple example is given
in Figure 2.9(a).
35
To avoid the dynamic meshes in an analysis from being affected by one specific
direction than the other two of coarsening, the AH-adaptive FEA method dynamically
changes the sequence of coarsening directions for different time increments:
1. At the first (and 4th, 7th, 10th, · · ·) time increment, it starts with (local isopara-
metric) r1-direction. After this direction is done, r2- and then r3-directions are
processed (see Figure 2.9(b)).
2. At the second (and 5th, 8th, 11th, · · ·) time increment, the sequence is r2- → r3-
→ r1-directions.
3. At the third (and 6th, 9th, 12th, · · ·) time increment: r3- → r1- → r2-directions.
Upon finishing coarsening the elements which are suggested to have larger element
sizes, the scheme performs element refining.
2.7 Element Refining
2.7.1 Creating Entities
Figure 2.10 illustrates the refining of an element in the three possible directions
(compared with the isotropic refinement scheme, which must generate eight elements and
the associated nodes at one time). Anisotropic refinement in any direction generates two
new elements and deactivates the original element, resulting in four new nodes (unless
some node has been defined at the same coordinates because of other existing elements).
Refined from an element in the rn direction, the generation and index numbers
which the two new elements, a and b, possess are defined by:
36
r1
r2r3
(-1,-1,0)
(-1,-1,0)
(-1,-1,0)
(0,0,0)
(-1,0,0)
(-1,-1,0)
(0,0,0)
(0,-1,0)
(-1,-1,0)(0,0,0)
r1-direction first
r2-direction first
(A,B,C) : the remaining generations to remesh (coarsen) for an element
(a) (b)
r1
r2 r3
Fig. 2.9. (a) Different new meshes using different sequences, (b) coarsening sequenceloop.
37
r1
r2r3
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Node 1Node 2
Node 5
Node 7 Node 8
Node 3 Node 4
Node 6
Node 1Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Node 1Node 2
Node 3
Node 5
Node 8Node 7
Node 6
Node 4Node 1
Node 2
Node 3 Node 4
Node 5Node 6
Node 7 Node 8
Node 1Node 2
Node 3
Node 5
Node 7 Node 8
Node 6
Node 4 Node 1Node 2dode 2de 2NodNododdN dN d 22dd
Node 3 Node 4Node 5
NodeNooode 6ode 6odeodeoode
Node 7 Node 8
r1-refinement
r2-refinement
r3-refinement
(a)
(b)
(c)
Isotropic
refinement
( (
(1)(2)
(3) (4)
(5)(6)
(7) (8)
(9)
(10)
(11)
(12)
(1)(2)
(3)
(4)
(5)(6)
(7) (8)
(9)(10)
(11)(12)
(1)(2)
(3)
(4)
(5)
(6)
(7)(8)
(9)
(10)
(11) (12)
Fig. 2.10. Refining an element in (a) r1- (b) r2- (c) r3- direction, and the node arrange-ment.
38
Iak = Ibk
Gak = Gbk
for k = 1, 2, 3, k �= n (2.25)
Ian = 2 ∗ Iorigin,n − 1
Ibn = 2 ∗ Iorigin,n
(2.26)
Gan = Gbn = Gorigin,n + 1 (2.27)
2.7.2 Transferring Solution Field and Boundary Conditions
Section 2.6.2 introduced the different categories of data to transfer. For a refine-
ment process in an AH-adaptive thermal analysis, the details for handling the quantities
are described below:
1) Node-wise quantities: As an example, take r1-refinement in Figure 2.10(a).
While the temperatures on the created nodes (9)–(12) could be interpolated as the
average of the corresponding two end-point on the edge, e.g.
T9 =12(T1 + T2)
if another node has been defined at the same coordinates by an adjacent element,
a duplicate node will not be created and the analysis skips the procedure for that specific
node.
2) Surface-wise quantities: The AH-adaptive scheme refines an element in one
local direction at a time. So, depending on the type of refinement, a surface on the
39
original element may either be split onto the two refined elements, or will remain a whole
piece on one of those if applicable. The boundary conditions are transferred accordingly.
3) Element-wise quantities:
1. Boundary condition: for either r1-, r2- or r3-refinement, if the original element
possesses an element-wise boundary condition, both of the two new elements will
be defined with the same quantity as well.
2. Remaining generations to refine/coarsen:
if an element is refined in rn direction, with (R1, R2, R3), where R′s are how many
generations for this element to re-mesh, then
RAi = RBi =
Ri if i = 1 − 3 and i �= n
Ri − 1 if i = n
(2.28)
where RA′s = RB
′s are the remaining generations of elements A and B to be
refined/coarsened.
2.7.3 Sequence of Element Refining
Because each refinement only involves the certain element itself which is going to
refine, unlike the effect of different coarsening sequences (Figure 2.9(a)), the sequence
of directions for refining elements does not influence the resulted mesh. Therefore, the
AH-adaptive scheme applies the same sequence: first r1, then r2, and finally r3 for every
time increment.
40
2.8 Identification of Dependent Nodes
Dependent nodes occur wherever adjacent elements have different element gener-
ations. Figure 2.11(B) shows an example mesh containing dependent (or constrained)
nodes in the AH-adaptive analysis. The solutions of the DOFs on the dependent nodes
will be constrained as they need to satisfy the linear, for hex8 elements, solution field
distributions for the adjacent elements. To account for the constrained DOFs, the con-
densation and recovery method [28] is applied.
2.8.1 Condensation and Recovery Theory
For a non-linear system, the incremental nodal solution is computed from the
algebraic system
Aδu = b (2.29)
where A is the tangent matrix with b being the negative of the residual, and δu
is the incremental solution during an iteration.
For each iteration, the system (Equation (2.29)) is processed by partitioning the
DOFs into
{δu} =
δur
δuc
(2.30)
where the subscript r stands for “retained”, and c stands for
41
“condensed”. Thus, δur represents the actual DOFs to be retained, and δuc
represents the condensed DOFs of the dependent nodes. Thus, the entire partitioned
non-linear system can be represented as:
Arr Arc
Acr Acc
δur
δuc
=
br
bc
(2.31)
The general representation of the constraint equations is given by
[Cr Cc
]
δur
δuc
={
Q
}(2.32)
where Cr and Cc are the coefficients for the retained and condensed nodes, respec-
tively, and Q is the constant in the system of constrained equations. For the AH-adaptive
analysis scheme, a constraint equation must have the form of
uc =N∑
k=1Ckuk (2.33)
where uc is the solution of the constrained node. N is the number of nodes on
which it depends, and Ck is the corresponding constraint coefficient.
The RHS terms in Equation (2.33) can be moved to the LHS of the equation:
42
b
1
2
x
y
3
a
em
em
men
meme
em
4
6
c
d
z
em
men
5
t
t
en
em
en
em
(A) (B)
Fig. 2.11. Mesh containing dependent nodes.
43
uc −N∑
k=1Ckuk = 0 (2.34)
So that the constant term Q in Equation (2.32) must be zero.
Utilizing the equations representing the constraints, we now have:
[Cr Cc
]
δur
δuc
={
0
}(2.35)
So,
{δuc
}= −[ Cc ]−1[ Cr ]
{δur
}= [ Crc ]
{δur
}(2.36)
where Crc = −C−1c Cr is the combined coefficient matrix.
By substituting Equation (2.36) into Equation (2.31),
[Arr + ArcCrc + CT
rcAcr + CTrcAccCrc
] {δur
}=
{br + CT
rcbc
}(2.37)
44
For example, in Figure 2.11, with 24 independent nodes and 4 constrained nodes,
the constraint equations for the dependent nodes are:
ua =12(u1 + u2) (2.38)
ub =12(u2 + u3) (2.39)
uc =12(u4 + u5) (2.40)
ud =12(u5 + u6) (2.41)
Equation (2.35) for this example system becomes
45
1 0 0 0 −12 −1
2 0 0 0 0 0 ... 0
0 1 0 0 0 −12 −1
2 0 0 0 0 ... 0
0 0 1 0 0 0 0 −12 −1
2 0 0 ... 0
0 0 0 1 0 0 0 0 −12 −1
2 0 ... 0
4∗28
ua
ub
uc
ud
u1
u2
u3
u4
u5
u6
u7
...
u24
28∗1
=
0
0
0
0
4∗1
(2.42)
Thus,
46
[Cc
]=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
= [ I ]4∗4 (2.43)
[Crc
]= −[ Cr ] =
12
12 0 0 0 0 0 ... 0
0 12
12 0 0 0 0 ... 0
0 0 0 12
12 0 0 ... 0
0 0 0 0 12
12 0 ... 0
4∗24
(2.44)
Therefore, all the matrices in Equation (2.37) can be acquired. Applying the
condensation theory in solving the system containing dependent nodes offers several
advantages over some other strategies, such as using Lagrange multipliers [28]. The
comparison can be seen in a future paper [29].
Practically, a constrained DOF is not necessarily at the average magnitude of
the other two DOFs like in Equation (2.38) – Equation (2.41). The following sections
will illustrate how to obtain constrained equations correspondingly, and what effects the
condensation brings to the entire system.
47
2.9 Pre-processing for the Condensed System
2.9.1 Determination of Constraint Equations
In Figure 2.12, nodes 1–10 are constrained by the corner nodes A–D on the surface
of element a, because only linear interpolations will be allowed on that surface. By using
the corresponding iso-parametric coordinates (r1 and r2), every constrained node may
be treated as depending on the four corner nodes of the surface. Hence the constraint
equation for a dependent node is:
Td =(1 − r1)(1 − r2)
4TA+
(1 + r1)(1 − r2)4
TB+(1 + r1)(1 + r2)
4TC+
(1 − r1)(1 + r2)4
TD
(2.45)
where r1 and r2 are the iso-parametric coordinates of the constrained node on
the surface. Td is the temperature of the dependent node. TA, TB , TC , TD are the
temperatures on the four corner nodes forming the element surface.
For a dependent node located on an edge of an interface, Equation (2.45) will
reduce to two terms only, which correspond to the fact that this node is actually con-
strained by the two end points of the edge. (For instance, in Figure 2.12 nodes 1 and 2
only depend on nodes A and B, while nodes C and D do not affect them at all.) On this
kind of nodes, the equations are in the form of
Td = αTe1 + βTe2 (2.46)
48
where α and β are the coefficients corresponding to two edge nodes e1 and e2
respectively, from Equation (2.45).
After acquiring all the constrained equations, the original system needs to be con-
densed. The effects on the system brought by the condensation procedure are introduced
and illustrated in the following sections.
2.9.2 Nonzero Fill-ins in Sparse Tangent Matrix for AH-Adaptive Mesh
The current research on the AH-adaptive analysis utilizes the IBM WSMP solver
[30], a sparse matrix solver which only utilizes information from the nonzero components
and saves it into linear arrays in order to reduce computational overhead. The nonzero
enforcement effect, which will be illustrated in this section, for the condensed matrix is
essential because it is necessary to determine the exact location of all nonzero positions
within the matrix in order to properly use the solver.
The condensation method (Section 2.8.1) actually can be perceived as splitting
the columns and rows of the constrained DOFs into the columns and rows of the DOFs
on which they are dependent. Figures 2.13, 2.14 and 2.15 demonstrate this effect with
an example.
1. On the basis of the mesh in Figure 2.13(a), the DOFs on nodes 2 and 11 need to
be constrained. Figure 2.13(b) shows the size of the original tangent matrix, where
the Ki,j ’s are the components, and the arrows indicate where the split rows and
columns should go upon being condensed.
49
A
Element a
B
C
D
AB
C D
12
3456
789
10
r1
r2
Fig. 2.12. Constrained nodes between adjacent elements.
50
2. The actual nonzero components in the original tangent matrix are presented in
Figure 2.14. Splitting the columns and rows of the constrained DOFs in the matrix
induces nonzero components at some positions which are originally zeros.
3. Figure 2.15 shows the condensed tangent matrix, and all the nonzero components
including those caused by the condensation.
Note that because this example mesh contains only three elements and sixteen
nodes, the non-zeros appear very dense in the tangent matrix (Figures 2.14 and 2.15).
Practical structures contain more elements and DOFs so that the tangent matrix is still
sparse.
2.9.3 Residual Array
Similar to the tangent matrix, the original residual array also requires to be con-
densed. However, the effect of condensing the residual array is simpler. Corresponding
to the same system in the above section for illustrating the tangent matrix, condensing
a residual array is shown in Figure 2.13(c).
2.10 Recovering the Condensed DOFs
After the system containing only the unconstrained (retained) DOFs has been
solved, the solutions for the remaining constrained DOFs can be recovered by applying
the constraint equations Equation (2.35). With the recovered magnitudes, the entire
solutions for all entities are acquired.
51
(a)
1
2
12
7
3
4
5
6
8
9
10
11
13
14
15
16
(b)
1 2 3 4 .....
1
2
3
4
.....
.....
.....
.....
.....
.....
.....
.....
.....
K1,1
K2,1
K3,1
K4,1
K1,2
K2,2
K3,2
K4,2
K1,3
K2,3
K3,3
K4,3
K1,4
K2,4
K3,4
K4,4
(c)
1
2
3
4
.....
16
.....
R1
R2
R3
R4
R16
Tangent Matrix Residual Array
.....
.....
.....
K10,1
K11,1
K12,1
K10,2
K11,2
K12,2
K10,3
K11,3
K12,3
K10,4
K11,4
K12,4
.....
16 .....
.....
.....
.....
.....
K16,1 K16,2 K16,3 K16,4
10
11
12
..... 16
.....
.....
.....
.....
.....
.....
K1,16
K2,16
K3,16
K4,16
.....
.....
.....
K10,16
K11,16
K12,16
.....
.........
.
K16,16
10 11 12
.....
.....
.....
K1,10
K2,10
K3,10
K4,10
K1,11
K2,11
K3,11
K4,11
K1,12
K2,12
K3,12
K4,12
K10,10
K11,10
K12,10
K10,11
K11,11
K12,11
K10,12
K11,12
K12,12
.....
.....
.....
K16,10 K16,11 K16,12
.....
.....
10
11
12
R10
R11
R12
.....
.....
x coefficient*
x coefficient*
x coefficient*
x coefficient*
x coefficient* x coefficient*x coefficient* x coefficient*
x coefficient*
x coefficient*
x coefficient*
x coefficient*
* The coefficient corresponding to the constrained equation
Fig. 2.13. Splitting the row and column of the dependent DOF.
52
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1
2
3
4
5
6
78
9
10
11
12
13
14
: nonzero elements
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
15 16
15
16
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Original tangent matrix for the system of equations
Fig. 2.14. Nonzero fill-in effect induced by condensing the tangent matrix.
53
1 73 4 5 6 8 10 1512 13 14
1
3
4
5
6
8
10
12
13
14
: original nonzeros in the stiffness matrix
: nonzero enforcements (fill-ins) due to the condensation effect
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Condensed tangent matrix
9 16
7
15
9
16
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Fig. 2.15. Nonzero fill-ins in the condensed matrix.
54
Chapter 3
Thermal Analyses Numerical Examples
3.1 Heat input model
Goldak’s double ellipsoid heat source model [31] is most commonly used for rep-
resenting the welding heat input. The model formulation is as stated below:
Q(x, y, z, t) =6√
3fqη
abcπ√
πe−3(x
2
a2 +y2
b2+z
2
c2)
(3.1)
where q is the welding heat input, η is the weld efficiency, x, y, and z are the
local coordinates of the heat source model, a is the weld width, b is the weld penetration
depth, c is the weld ellipsoid length. The parameter f and c is dependent upon the weld
type.
3.2 Linear Weld Path — Comparison of Static and AH-Adaptive Anal-
ysis
In order to 1) evaluate the computational cost using both the static and the
AH-adaptive schemes, and 2) verify that the adaptive analysis results match the static
analysis results, comparisons between the two analyses are performed.
A laser weld on a plate as shown in Figure 3.1 is simulated, with a travel speed
of 12.7 mm/s, a delivered power of 4.6 kW, and an efficiency of 0.5. The plate has
55
dimensions of 152.4 mm in the x-, and 304.8 mm in the y- directions, with either 10 mm
or 5 mm thickness in the z-direction. Material properties for ASTM 131 grade EH-36
steel are used [32]. Eight-node hexagonal brick-type elements (hex8) are utilized in the
analyses.
3.2.1 Hardware
The simulations were performed on a SGI altix 350 system with 8 CPUs.
3.2.2 Software
The software used in this study is an in-house finite element code written in
Fortran 90. An implicit solution scheme and the Newton-Raphson method were used
to solve the non-linear problems in an iterative fashion. Most procedures are written to
be sequential, except the solver which uses all available CPUs for parallel computing.
However, the resulting peak number of DOFs in the AH-adaptive examples is small, so
that the equation solver executes sequentially as well.
3.2.3 Analysis Results
The mesh used in the static simulation is illustrated in Figure 3.1, with the thermal
simulation results shown in both Figures 3.2, a 3-D structural view, and 3.3, a cross-
section view. Meanwhile, Figure 3.4 shows the starting (initial) mesh used in the AH-
adaptive analysis. The dynamic mesh at an instant when the heat source has traversed
approximately three-quarters of the way across the plate is shown in Figure 3.5. The
results, which closely match the static analysis result, are shown in Figures 3.6–3.8.
56
Figure 3.6 shows the temperature fringes with the peak temperature of 1730◦C.
Compared to the peak temperature in the static mesh analysis (Figure 3.2) which is
1700◦C, the error is about 1.8 %. Also, Figure 3.7 provides a zoom-in view showing the
high temperature area and the detailed dynamic mesh more clearly. The cross-section
view in Figure 3.8 furthermore gives similar temperature distributions to those from the
static mesh analysis.
3.2.4 Comparison
Table 3.1 gives the computation statistics for both the static and the AH-adaptive
analyses. Note that the presented examples demonstrate the ability to reduce the analysis
CPU time by 86.95 % compared to the conventional static solution.
3.3 Combined Weld Path (Curved and Linear) — Evaluation of AH-
Adaptive Analysis Scheme
The AH-adaptive FE scheme is also applied to simulate a welding procedure
on a 3ft × 3ft plate shown in Figure 3.9. Note also the heat source does not merely
move in linear paths, but also moves along a one-quarter arc of a circular path. In this
case, the heat source is a hybrid of laser-GMAW weld, with a gap of 1/16 in, a travel
speed of 8.47 mm/s, a delivered power of 4.3 kW, and an efficiency of 0.6. The plate has
dimensions of 914.4 mm in the x- and y- directions, with either 10 mm or 5 mm thickness
in the z-direction. The material properties, hardware and software are the same as those
described in Section 3.2.
57
X
Y
Z
X
Y
Z
Fig. 3.1. Static mesh.
58
X
Y
Z
1.70+03
1.80+03
1.68+03
1.56+03
1.44+03
1.32+03
1.20+03
1.08+03
9.60+02
8.40+02
7.20+02
6.00+02
4.80+02
3.60+02
2.40+02
1.20+02
0. default_Fringe :Max 1.70+03 @Nd 19820Min 1.94+01 @Nd 27006
X
Y
Z
Fig. 3.2. Temperature result (◦C) for the static mesh analysis.
59
X Y
Z
1.76+03
1.80+03
1.68+03
1.56+03
1.44+03
1.32+03
1.20+03
1.08+03
9.60+02
8.40+02
7.20+02
6.00+02
4.80+02
3.60+02
2.40+02
1.20+02
0. default_Fringe :Max 1.76+03 @Nd 19782Min 1.94+01 @Nd 27006
X Y
Z
Fig. 3.3. Cross section view for the static analysis result (◦C).
60
X
Y
Z
X
Y
Z
Fig. 3.4. Initial mesh for the AH-adaptive analysis.
61
X
Y
Z
X
Y
Z
Fig. 3.5. Adaptive mesh at the instance of comparison.
62
X
Y
Z
-6.71+01
1.80+03
1.68+03
1.56+03
1.44+03
1.32+03
1.20+03
1.08+03
9.60+02
8.40+02
7.20+02
6.00+02
4.80+02
3.60+02
2.40+02
1.20+02
0. default_Fringe :Max 1.73+03 @Nd 57016Min -6.71+01 @Nd 135
X
Y
Z
Fig. 3.6. Temperature result of the AH-adaptive analysis (◦C).
63
X
Y
Z
1.80+03
1.68+03
1.56+03
1.44+03
1.32+03
1.20+03
1.08+03
9.60+02
8.40+02
7.20+02
6.00+02
4.80+02
3.60+02
2.40+02
1.20+02
0. default_Fringe :Max 1.73+03 @Nd 57016Min -6.71+01 @Nd 135
X
Y
Z
Fig. 3.7. Zoom-in view of the AH-adaptive analysis result (◦C).
64
X Y
Z
1.79+03
-5.90+01
1.80+03
1.68+03
1.56+03
1.44+03
1.32+03
1.20+03
1.08+03
9.60+02
8.40+02
7.20+02
6.00+02
4.80+02
3.60+02
2.40+02
1.20+02
0. default_Fringe :Max 1.79+03 @Nd 57005Min -5.90+01 @Nd 57923
X Y
Z
Fig. 3.8. Cross section view of the AH-adaptive analysis result (◦C).
65
Table 3.1. Comparison between the static and the AH-adaptive analyses on the modelStatic Mesh Analysis AH-Adaptive Analysis
Number of Nodes Statically 77979 Initial 240 Peak 6713Average 2085(Total generated 77584)
Number of Elements Statically 70800 Initial 120 Peak 5078Total Number ofTime Increments
189 154
Peak Temperature(◦C)
1760 1790
Total Analysis CPUTime
3378 sec 440.71 sec
Processing adaptivemesh information
(N/A) 258.4 sec
Residual assembling 1098 sec 36 secTangent matrix as-sembling
1186 sec 47 sec
Algebraic equationsolving
905 sec 80 sec
66
3.3.1 Analysis Results
Two sets of analyses are performed as listed in Table 3.2 to demonstrate the effects
of using different sizes of the moving forced refinements on CPU usage.
The temperature results and the generated meshes at several time increments for
case L1 are illustrated in Figures 3.10 to 3.17. The re-meshing control factors are defined
as follows:
• Radii of enforced refinements:
1. inner 12.14 mm within which elements are refined to the finest allowable
generations, and
2. outer 13.11 mm within which elements are forced to be refined to
Max(Int(0.5 ∗ Nmax), Ngrad)
where Max() is taking the maximum value, Int() is taking the integer value,
Nmax is the maximum allowable generation determined by the minimum
element length, and Ngrad is the calcualtion result of element generation
evaluated from the gradient measure (Section 2.4).
• Minimum length of elements: 0.6 mm.
• Permissible gradient: 20.0◦C/(isoparametric length unit).
Note that the outer radius does not necessarily have to be much larger than the
inner radius for an analysis, because as long as the inner radius can generate sufficient
67
Table 3.2. Statistics of the AH-adaptive analysisCase number L1 L2Radii of EnforcedRefinement
12.14mm and 13.11mm 13.04mm and 14.14mm
Minimum Length ofElements
0.6mm 0.6mm
Permissible Gradient 20.0 20.0Total Number ofTime Increments
1144 1174
Number of Nodes Initial 440 Peak 14438 Initial 440 Peak 18502Number of Elements Initial 288 Peak 11193 Initial 299 Peak 13518Average number ofnodes per iteration
7817 10295
Total Analysis CPUTime
9050 sec 12539 sec
Residual Assembling 501 sec 708 secTangent Matrix As-sembling
623 sec 901 sec
Algebraic EquationSolving
1008 sec 1615 sec
Adaptive Mesh In-formation Processing
6906 sec 9015 sec
68
elements to integrate the heat source(s), the outer radius functions just as the regions of
transition elements. Figures 3.10 and 3.11 are from the instant t = 15.4 sec, when the
heat source is on the first linear path segment. Figure 3.12 shows the time increment
t = 40.4 sec when the heat source starts to enter the circular path, with Figure 3.13
providing a zoom-in view to show more details of the mesh and the temperature results.
Furthermore, Figure 3.14 demonstrates the results when the heat source is about leaving
the 3ft × 3ft plate (t = 100 sec). While a zoom-in effect for the details can be seen in
Figure 3.15, Figure 3.16 shows how the temperature and the dynamic mesh are through
the thickness direction at this instant. In addition, Figure 3.17 which has exactly the
same mesh density as the initial mesh, is the temperature results upon cooling when t
= 3600 sec.
3.3.2 Comparison to static mesh analysis
Analyzing the 3ft × 3ft plate using a conventional static mesh is intractable due
to:
• the large number of elements and DOFs which would be required to adequately
describe the structure.
• the complexity of manually generating transition elements between the high ele-
ment density areas and elsewhere, especially near the circular weld path.
To generate a static mesh for the 3ft × 3ft plate (of the same size shown in Figure
3.9), the estimated number of nodes is calculated from the static mesh of Figure 3.1
used in Section 3.2. The small structure in Figure 3.1 is cut from the 3ft × 3ft plate,
69
X
Y
Z X
Y
ZX
Y
Z X
Y
Z
Fig. 3.9. The plate and the initial mesh.
70
X
Y
Z
1.95+03
1.82+03
1.69+03
1.56+03
1.43+03
1.30+03
1.17+03
1.05+03
9.17+02
7.89+02
6.60+02
5.32+02
4.03+02
2.75+02
1.46+02
1.77+01 default_Fringe :Max 1.95+03 @Nd 2792Min 1.77+01 @Nd 469
X
Y
Z
Fig. 3.10. Temperature result (◦C) at t = 15.4 sec.
71
X
Y
Z
1.95+03
1.82+03
1.69+03
1.56+03
1.43+03
1.30+03
1.17+03
1.05+03
9.17+02
7.89+02
6.60+02
5.32+02
4.03+02
2.75+02
1.46+02
1.77+01 default_Fringe :Max 1.95+03 @Nd 2792Min 1.77+01 @Nd 469
X
Y
Z
Fig. 3.11. Zoom-in of the temperature result (◦C) at t = 15.4 sec.
72
X
Y
Z
2.03+03
1.90+03
1.76+03
1.63+03
1.49+03
1.36+03
1.22+03
1.09+03
9.50+02
8.15+02
6.79+02
5.44+02
4.08+02
2.73+02
1.37+02
1.91+00 default_Fringe :Max 2.03+03 @Nd 7022Min 1.91+00 @Nd 300
X
Y
Z
Fig. 3.12. Temperature result (◦C) at t = 40.4 sec.
73
X
Y
Z
2.03+03
1.90+03
1.76+03
1.63+03
1.49+03
1.36+03
1.22+03
1.09+03
9.50+02
8.15+02
6.79+02
5.44+02
4.08+02
2.73+02
1.37+02
1.91+00 default_Fringe :Max 2.03+03 @Nd 7022Min 1.91+00 @Nd 300
X
Y
Z
Fig. 3.13. Zoom-in of the temperature result (◦C) at t = 40.4 sec.
74
X
Y
Z
1.67+03
1.56+03
1.45+03
1.34+03
1.23+03
1.12+03
1.01+03
8.99+02
7.88+02
6.78+02
5.67+02
4.56+02
3.46+02
2.35+02
1.24+02
1.34+01 default_Fringe :Max 1.67+03 @Nd 3258Min 1.34+01 @Nd 87
X
Y
Z
Fig. 3.14. Temperature result (◦C) at t = 100 sec.
75
X
Y
Z
1.67+03
1.56+03
1.45+03
1.34+03
1.23+03
1.12+03
1.01+03
8.99+02
7.88+02
6.78+02
5.67+02
4.56+02
3.46+02
2.35+02
1.24+02
1.34+01 default_Fringe :Max 1.67+03 @Nd 3258Min 1.34+01 @Nd 87
X
Y
Z
Fig. 3.15. Zoom-in of the temperature result (◦C) at t = 100 sec.
76
X
Y
Z
1.67+03
1.56+03
1.45+03
1.34+03
1.23+03
1.12+03
1.01+03
8.99+02
7.88+02
6.78+02
5.67+02
4.56+02
3.46+02
2.35+02
1.24+02
1.34+01 default_Fringe :Max 1.67+03 @Nd 3258Min 1.34+01 @Nd 87
X
Y
Z
Fig. 3.16. Cross-section view from the side at t = 100 sec (◦C).
77
X
Y
Z
6.11+01
5.90+01
5.68+01
5.46+01
5.24+01
5.02+01
4.80+01
4.59+01
4.37+01
4.15+01
3.93+01
3.71+01
3.50+01
3.28+01
3.06+01
2.84+01 default_Fringe :Max 6.11+01 @Nd 409Min 2.84+01 @Nd 25
X
Y
Z
Fig. 3.17. Temperature result (◦C) at t = 3600 sec.
78
and contains only 152.4 mm of the linear weld path portion. Its width is also only 304.8
mm (less than the width 914.4 mm of the entire plate). The total weld length on the
3ft × 3ft plate is 609.6 mm (two linear segments) + 215.8 mm (circular path) = 825.4
mm. Therefore, considering there are 77979 nodes on the mesh of Figure 3.1, using
similar mesh densities for the 3ft × 3ft plate would result to an estimated number of
nodes similar to 77979/152.4*825.4=422335. Ref. [33] performed a similar analysis of a
model with 424343 DOFs. The thermal analysis was performed on different computers
resulting in the following run times:
1. On a 16-CPU Unisys ES7000 system for the entire simulation, which consisted of
1579 time increments, the analysis was finished within 56 hours of wallclock time,
resulting to a 56*3600/1579=127.68 (wallclock) sec/time-increment.
2. On the 8-CPU SGI altix 350 system, which is the same equipment (Section 3.2.1)
used to perform all AH-adaptive analysis examples in this work, only for the first
38 increments and using only 1 CPU, the partial analysis took 81302 sec to finish,
and it results to a 81302/38=2139.5 sec/time-increment.
The AH-adaptive analyses of the 3ft × 3ft plate resulted to total CPU usages per
time increment are 9050/1144=7.91 sec and 12539/1174=10.68 sec for cases L1 and L2
respectively using one CPU of the SGI altix 350 sustem. The CPU time reduction against
the static mesh is thus 100%×(1−7.91/2139.5) = 99.63% and 100%×(1−10.68/2139.5) =
99.5% for analysis cases L1 and L2 respectively. If the necessary DOFs for a static mesh
analysis is larger, the efficiency of the adaptive methodology is expected to be even
79
much higher because the computational cost of decomposing the tangent matrices will
be reduced more significantly.
3.3.3 CPU scaling with model size
Using a standard direct sparse solver, the computational cost to factorize and
solve the equation system is reported to scale as O(n2) for 3D simulations (and O(n32 )
for 2D simulations) [34], where n is the total number of equations in the system. How-
ever, benefitting from the large reduction of DOFs, in the AH-adaptive analysis the
equation solving is not dominating the total computation. Furthermore the adaptivity
overhead induced by processing the adaptive mesh information is designed to be linear
(O(n)) which is confirmed by comparing the CPU usage for adaptive processing per time
increment per average number of nodes for all adaptive results:
• From the AH-adaptive analysis of the small model, which has average number
of nodes of 2085, as listed in Table 3.1 the CPU usage spent on adaptive mesh
information per time increment per average number of nodes is 258.4/154/2085 =
0.8048 × 10−3 sec.
• From the first set of the AH-adaptive analysis in Table 3.2, which has an average
number of nodes of 7817, the adaptive mesh information processing CPU usage per
time increment per average number of nodes is 6906/1144/7817 = 0.7723 × 10−3
sec.
80
• The second set in Table 3.2 has an average number of nodes of 10295, and the
adaptive mesh information processing per time increment per average number of
nodes is 9015/1174/10295 = 0.746 × 10−3 sec.
81
Chapter 4
The AH-Adaptive Mechanical Analysis Scheme
For a mechanical analysis, the algorithm of the created AH-adaptive FE scheme
is shown by the flow chart in Figure 4.1. While some procedures are similar to the
counterparts in the AH-adaptive thermal analysis [35], they differ due to either expanded
number of DOFs per node, or different quantities. Furthermore, because of a mechanical
analysis tending to diverge more easily then a thermal analysis (among the reasons are
such as plasticity, large deformation theory.), additional procedures are implemented.
For illustrative purposes the following sections are based on hex8 elements wherever a
specific element type is necessary.
4.1 Governing Equations for Quasi-Static Structural Analysis
An ordinary FEA approach on quasi-static mechanical analysis is to construct
the system of equations to solve for the displacement response (elasto-plasticity) :
The stress equilibrium equation is given as follows:
∇rσ(r, t) + b(r, t) = 0 in volumn V (4.1)
where sigma is the stress, and b is the body force.
82
Read modeldata file
Read input control file
Initialization of informationarrays
inc = 1
Elementcoarsening
Elementrefining
Identification ofdependent nodes
Pre-processingfor the condensedsystem
iter = 0
Assemble theresidual andstiffness
Solve thesystem
Update thesolution vector
Recovering thecondensedDOFs
e
n
iter = iter + 1
the procedures utilized in ordinary FEA (static mesh)
the procedures for AH-adaptive analysis ability
Gradientmeasure calculations
Evaluation ofrefinementlevel
MovingForcedRefinement
control criteria for generating new elements
Gauss point quantitiesbalancing
s
Readtemperatureresults
if eps(L2 norm ofincrementalsolution)< epslim<
No (not conve
Yes (con
If exceeds the maximun # of iteration
No
Yes (Cons
Acquire thesecondaryquantities
if time <maxtimemaxtime
inc = inc + 1
Analysisfinished
nvergent)
No
Yes
erging yet)
sidered divergent)
Re-calling thepreviously convergent mesh(connectivities, solutions)
If too many cutbacks
y
Cutback with a smaller time step
No
Yes
Keep the same adaptive mesh
Cut the time step
Save this convergentmesh(connectivities, solutions)
convergence efficiency improving strategy
Fig. 4.1. Flow chart of the AH-adaptive FE analysis scheme.
83
4.1.1 Small Deformation
In small deformation theory, the total strain ε is Green’s strain.
ε(r, t) =12
[∇ru(r, t) +
(∇ru(r, t)
)T]
(4.2)
Assuming small deformation thermo-elasto-plasticity, the total strain ε can be
decomposed into three terms:
ε = εe + εp + εt (4.3)
where εe, εp and εt are the elastic strain, plastic strain and thermal strain, respectively.
Using the above equation, the stress strain constitution relationship is
σ = C(ε − εp − εt) (4.4)
where C is the material stiffness matrix.
Applying the associative J2 plasticity [36], the yield function f is
f = σm
− σY
(εq, T ) (4.5)
where σm
and σY
are the Mises stress and yield stress.
Active yielding occurs when f ≥ 0. The evolution of εq
for active yielding can be
evaluated by the radial return algorithm [37], and then ε̇p
can be calculated from
ε̇p
= ε̇qa (4.6)
84
where a is the flow vector. The initial and boundary conditions can be found in [13].
4.1.2 Large Deformation
Illustrated in [38] is the large deformation theory:
For Total Lagrangian formulation, the basic equation is
∫V 0
St+�t
0,ijδε
t+�t
0,ijdV
0 = Rt+�t (4.7)
For Updated Lagrangian formulation, the basic equation is
∫V t
St+�t
t,ijδε
t+�t
t,ijdV
t = Rt+�t (4.8)
where
V0 is the volume at time 0
Vt is the volume at time t
St+�t
0,ij, S
t+�t
t,ijare the second Piola-Kirchhoff stress tensor
δεt+�t
0,ij, δε
t+�t
t,ijare the incremental Green-Lagrange strain tensor
Rt+�t is the external virtual work
An AH-adaptive mechanical analysis involves the tasks illustrated in the following
sections.
85
4.2 Initialization of Information Arrays
Throughout an entire analysis, the re-meshing procedures generate new entities.
Arrays which save this information associated with nodes and elements are necessary in
order to transfer the properties between the entities and properly construct the new mesh.
Figures 4.2 and 4.3 show the structures of the arrays. In mechanical analyses, element
arrays include Gauss point quantities (Figure 4.3), such as stresses and strains, which
are not used in thermal analyses [35]. Other than this difference, analogous concepts on
these node-wise and element-wise arrays are illustrated in [35].
4.3 Control Criteria on Generating Elements for Self-Adapting Dy-
namic Meshes
At the beginning of an analysis, and between two time increments, an initial/current
adaptive mesh will be modified to generate a new adaptive mesh by refining or coarsening
the elements as necessary. The key controlling criteria are:
1. Gradient measures. These gradients are compared to the desired permissible gradi-
ent. Elements are refined when the gradient measures are too high so as to request
for smaller element sizes, and coarsened if the measures are too low.
2. Moving forced refinement, which are set to track the moving heat source, and force
refinement in order to guarantee sufficient element density to adequately integrate
the heat source.
86
Node 1 Node 2 ..... Node Nmax
The temporary array for node data during re-meshing
x-coordinate
Nmax : the maximum # of total temporary nodes allowed during one loop of re-meshing
# of previous mesh nodes .....
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
Node 1 Node 2 .....
x-coordinate
# of active nodesafter re-meshing
y-coordinate
z-coordinate
nodal solution
*node-wise boundary conditions
The array of active node data
(With the information on what elements are active)
The nodes on active elements
* : actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
Fig. 4.2. The information array of node data.
87
Whether it is active
The initial element it comes from
..... (1:Active)0
Emax : the maximum # of total temporary elements allowed during one loop of re-meshing
Element generations
Index number associated with the generations
Elem 1 Elem 2 ..... Elem Emax# of previous mesh elements .....
Surface-wise B.C.
Gauss point quantities
1 0 ..... 1
The temporary array for element data during re-meshing
Initial Elem 1 Intial Elem 2 ..... Initial Element N
Maximum generations to be refined(for all 3 local directions seperately)
The material group
Element type configuration
The base (reference) array for all initial elements
Elem 1 Elem 2 .....# of active elementsafter re-meshing
The array of active element data
: actual number of rows depends on how many different kinds of boundary conditions may be applied in the analysis
b
b
b
The initial element it comes from
Element generations
Index number associated with the generations
b
b
b
a
: contains the information rows for element type, number of nodes in an elementa
: contains 3 rows, for the information corresponding to r1,r2,r3-directionsb
d
c
c
d : contains rows for stresses, strains, equivalent plastic strain, etc.
Surface-wise B.C.
Gauss point quantitiesd
c
Remaining generationsto refine / coarsen
Remaining generationsto refine / coarsen
Fig. 4.3. The information array of element data.
88
4.4 Gradient Measure Definition
Except at the start of an analysis, the AH-adaptive scheme examines elements on
a current mesh after the solution field has been acquired. An element with a smoother
gradient field will tend to be given a larger element size through coarsening, while an
element with a higher gradient demands to be refined to a smaller size to better reflect
the steep gradient.
4.4.1 Review of isotropic norm definitions
For isotropic FE analyses, [25] applies error norm derivations which build on
concepts introduced in [26, 27]. The need to isotropically refine or coarsen the elements
in a mesh is calculated according to:
‖e(i)‖ ≤ Ce(i)
hp−m+d (4.9)
where e(i) denotes the error in an element i, h is the maximum element diagonal
length, p is the order of the shape function, m is the highest order of differentiation in
the strain-displacement relation, and d equals to 1, 2, or 3 depending on the number
of dimensions. A new element size is computed after the element local error norm is
normalized by an additionally evaluated global gradient field.
89
Though these norm equations serve for isotropic refinement and coarsening, an
anisotropic analysis requires calculations for gradients independently for all (local r1-,r2-
,r3-) directions. The gradient evaluation for the AH-adaptive FEA scheme is developed
as in the following section.
4.4.2 Gradient measures of AH-adaptive mechanical analysis
While in an AH-adaptive thermal analysis the gradient measures are evaluated
according to the temperatures in the previous mesh [35], in a mechanical analysis there
are many solution fields such as nodal displacements, element-wise plastic strains or
stresses, and nodal peak temperatures. These different quantities can be taken as the
basis of the gradient measures. The mechanical analyses which will be demonstrated are
structural responses simulations with inputs of temperature results (from an AH-adaptive
heat transfer analysis). Note that the AH-adaptive scheme can also be applied for a
pure mechanical simulation without temperature inputs, e.g. only subject to mechanical
forces.
Peak Temperature as the Gradient Measure
Take an example of using peak temperatures (the highest temperature a node of a
specific coordinates has experienced from the beginning up to a current time increment)
as the gradient measure. Consider that at a time increment, the peak temperatures
on the nodes of an element are expressed as Tp
= [Tp1, T
p2, ..., Tp8]T . Thus, the peak
temperature Tp
of any interior point in this element is calculated by
Tp
= N · Tp (4.10)
90
where N is the shape function. This leads to the gradient definitions of the
solution field in all directions as
∇rTp=
dTp
dr=
d(N · Tp)
dr=
dNdr
· Tp
(4.11)
where the local gradient ∇r is a 3 ∗ 1 vector.
The AH-adaptive scheme evaluates the local gradients in an element at the center
of isoparametric coordinates (r1, r2, r3 = 0). Therefore, the gradient measures G for all
three directions (three scalars expressed in a vector form) are:
G = [Gr1
, Gr2
, Gr3
]T = |∇rTp|centroid
= |(dNdr
)centroid
· Tp| (4.12)
where | | denotes the absolute values.
Stresses as the Gradient Measure
If using Gauss point quantities such as stresses, the first step is to acquire the
stress field in an element. And then the gradients of stress field at the centroid can be
evaluated. For example, for a hex8 element that has 2 × 2 × 2 Gauss points, the stress
field within the element is
91
S = 0.577350269189626 × N · S (4.13)
where N is the shape function, S = [S1,S2, ...,S8]T is the array of Gauss point
magnitudes. And
∇rS =dS
dr=
d(N · S)dr
=dNdr
· S (4.14)
where the local gradient ∇r is a 3 ∗ 1 vector.
Therefore, the gradient measures G for all three directions (three scalars expressed
in a vector form) are:
G = [Gr1
, Gr2
, Gr3
]T = |∇rS|centroid= |(dN
dr)centroid
· S| (4.15)
where | | denotes the absolute values.
92
4.4.3 Evaluation of refinement level
For each specific simulation, a desired permissible gradient Gp
is set at the start
of the analysis. The need to refine or coarsen each element is determined by comparing
the magnitudes of the gradients:
Ri= log2
Gri
Gp
(4.16)
where Ri(i = 1, 2, 3) determines the need to refine or coarsen in each direction. If
Ri
is positive, the element requests to be refined in the specific direction. On the other
hand, a negative number of Ri
suggests to coarsen the element.
Among the benefits brought by these derived equations are:
1. they can be easily derived for either orthogonal or skewed elements (Figure 2.4),
2. the computed gradient measures are completely unaffected by the difference in
global dimension sizes among all elements. Globally normalizing the element-wise
magnitudes can be avoided,
3. these independent element-wise calculation makes it practical to shorten the real-
time analysis length by utilizing multi-CPU parallelization if desired.
93
4.5 Moving Forced Refinement
In the AH-adaptive analysis, a starting mesh can be set to be the minimum
element densities whenever and wherever the Gauss points are sufficient to integrate
the system energy, as this also benefits an analysis by saving more CPU usage. So the
starting mesh may be very coarse. However, because
• for the time increments when the Gauss points in such a coarse initial mesh can
be too distant from the highest energy concentration to sufficiently integrate the
heat source,
• a forward re-meshing technique is utilized to significantly reduce the computational
cost compared to iterative re-meshing techniques.
moving forced refinement within spherical regions moving along with the heat
source(s) (see Figure 2.5) is introduced to trigger and enforce sufficient element densities.
The dual, or multiple if needed, moving spheres guarantee different degrees of refinement.
The concepts and utilization of the moving forced refinement in an AH-adaptive
mechanical (structural) analysis is the same as that of an AH-adaptive heat transfer
(thermal) analysis. The detailed illustrations can be seen in the previous work [35].
4.6 Element Coarsening
For both thermal analyses and mechanical analyses, the AH-adaptive FE scheme
processes element coarsening before refining. This sequence reduces the number of tem-
porary nodes and elements than the opposite way. An element is only allowed to coarsen
94
with its pair element if 1) the element has the gradient measure suggesting a larger size,
and 2) the suggested coarsening can not go below the necessary generation enforced by
the moving spheres.
4.6.1 Mutually Coarsenable Elements
In the AH-adaptive scheme, coarsening does not depend on the sequence of how
elements were refined. Therefore, how elements are coarsened does not need to be bound
with (sometimes can even be “hampered” by) the refining history for any element. Thus,
there can be more flexibility in re-meshing. However, coarsening without any reference
could induce an undesirable situation, such as that illustrated in Figure 2.8(a1). Element
A can not be coarsened with element B or C no matter how much coarser it desires,
though the elements are of exactly the same generations in all directions. In the example
elements B, C can still coarsen mutually, however, this clearly will not help coarsening
element A. Note that the elements could be coarsened into a single element whenever
they need to, if they are coarsened in the proper manner (Figure 2.8(a2)).
To make elements able to appropriately coarsen together with their “pair” (or
can be referred to as “sibling”) elements, regardless of what element generations they
are, three index numbers (I1, I2, I3) corresponding to the three local directions, are
assigned to every element and associated with current element generations (G1, G2, G3).
The previous paper on thermal analyses [35] illustrates how to use these information to
accurately determine the mutually coarsenable elements.
95
4.6.2 Transferring Data
After each coarsening, the original two elements will be deactivated. Therefore,
the data on the old elements need to be transferred. In an AH-adaptive mechanical
analysis, the primary categories of data to transfer for the finite element entities are:
1. Node-wise quantities: nodal solutions from the previous time increment, and bound-
ary conditions such as prescribed displacements.
2. Surface-wise quantities: boundary conditions such as surface pressure.
3. Element-wise quantities: Gauss point quantities, and remaining generations to
refine/coarsen, etc.
4.6.3 Processing the Nodes and Elements
Element coarsening never creates new nodes. Neither does it involve immediate
transfer of nodal quantities. Also note that a node can not be deactivated simply because
of element coarsening, as an adjacent element may still be using the node. Eliminating
any node can only be operated when the re-meshing procedure is complete, if not any
element is using it at all.
For a new generated element, the surface- and element-wise quantities (Section
4.6.2) are processed as:
a) Surface-wise quantities: Properties on the surfaces of a new element will inherit
from those of the corresponding old element surfaces.
b) Element-wise quantities:
96
1. Remaining generations to refine/coarsen:
if two elements A and B are to be coarsened in rn
direction, and
element A: (RA1, R
A2, RA3)
element B: (RB1, R
B2, RB3), where R
A′s and R
B′s are the remaining genera-
tions to re-mesh in all three directions for elements A and B, respectively, the
corresponding numbers for the new element C are
RCi
=
Int(R
Ai+R
Bi2 ) if i = 1 − 3 and i �= n
Int(R
Ai+R
Bi2 ) + 1 if i = n
(4.17)
where RC′s are how many generations element C needs to be re-meshed, i corre-
sponds to local directions, and Int() is taking the integer value.
2. Gauss points quantities: the magnitudes on a new Gauss point will be interpolated
from old Gauss points. Figure 4.4(a) shows an example for a specific element
coarsening.
4.6.4 Sequence of Element Coarsening
The AH-adaptive scheme processes all coarsenable elements in one specific local
direction (e.g. r1-direction) first. After this certain direction is finished, a second local
direction and then the third direction are operated sequentially. The reasons why the
sequence of directions matter, and how the starting direction is determined are illustrated
in [35].
97
r1
r2r3
r1-coarsening
XX
XX
XX
XX
##
##
##
##
: Gauss points on the new elemnt
: Gauss points on 1st old element
: Gauss points on 2nd old element
X
#
X
X
(a)
r1-refining
XX
: Gauss points on the new elemnts
: Gauss points onthe old elementX
X
X
(b)
XX
XX
XX
Fig. 4.4. Gauss point interpolations for (a) coarsening, (b) refining
98
4.7 Element Refining
4.7.1 Creating Entities
Anisotropic refinement creates two new elements, and the original element need
to be deactivated. For examples consisting of hex8 elements, a maximum of four new
nodes can be generated. However, if an existing node used by other element(s) has been
defined at the same coordinates, duplicating nodes is not allowed in the AH-adaptive
scheme.
4.7.2 Index and Generation Numbers
The index and generation numbers for the two new elements a and b are:
If refined in rn
direction, then
Iak
= Ibk
Gak
= Gbk
for k = 1, 2, 3, k �= n (4.18)
Ian
= 2 ∗ Iorigin,n
− 1
Ibn
= 2 ∗ Iorigin,n
(4.19)
Gan
= Gbn
= Gorigin,n
+ 1 (4.20)
4.7.3 Transferring Solution Field and Boundary Conditions
For refining an element,
1) Node-wise quantities: If a new node is generated, e.g. node 9 in the middle of
two original nodes 1 and 2,
99
ui,9 =
12(u
i,1 + ui,2)
where i = 1–3 represents x-, y-, and z-direction respectively.
2) Surface-wise quantities: Surfaces of the original element may either be split
onto the two refined elements or remain a whole piece, depending on the refinement
direction. Boundary conditions are transferred accordingly.
3) Element-wise quantities:
1. Remaining generations to refine/coarsen:
if an element is refined in rn
direction, with (R1, R2, R3), where R′s are how many
generations for this element to re-mesh, then
RAi
= RBi
=
Ri
if i = 1 − 3 and i �= n
Ri− 1 if i = n
(4.21)
where RA′s = R
B′s are the remaining generations of elements A and B to be
refined/coarsened.
2. Gauss points quantities: On the basis of old Gauss points, The example of Figure
4.4(b) illustrates that magnitudes on new Gauss points are calculated through
either interpolation or extrapolation. coarsening.
100
4.7.4 Sequence of Element Refining
The difference of resulted meshes between various sequence of operating directions
(in Section 4.6.4, or [35]) does not occur at element refinements, because each refining
only involves self-dividing regardless of its neighboring elements.
4.8 Identification of Dependent Nodes
If adjacent elements have different element generations, dependent (also referred
to as “constrained”) nodes take place. Illustrations for the causes, and how to accurately
handle these nodes in the system of equations with condensation and recovery method
[28], can be seen in [35]. Note that the difference between a thermal and a mechanical
analysis in the AH-adaptive scheme is the number of DOFs per node. In a structural
analysis, all DOFs on a dependent node have to be constrained as well.
4.9 Pre-processing for the Condensed System
4.9.1 Determination of Constraint Equations
As more details illustrated in [35], each dependent node can be regarded to be
constrained by the four corner nodes of the surface on which it locates. Hence the
constraint equation is in the form of:
ui,d
=(1 − r1)(1 − r2)
4ui,A
+(1 + r1)(1 − r2)
4ui,B
+(1 + r1)(1 + r2)
4ui,C
+(1 − r1)(1 + r2)
4ui,D
(4.22)
101
where r1 and r2 are the iso-parametric coordinates of the constrained node on
the surface, i = 1–3 correspond to the three spatial dimensions, ud’s is the solutions
(displacements) of the dependent node, uA
’s, uB
’s, uC
’s, uD
’s are those on the four
corner nodes which form that element surface.
And if a dependent node locates on an edge of an interface, Equation (4.22) will
reduce to two terms only:
ui,d
= αui,e1 + βu
i,e2 (4.23)
where α and β are the coefficients corresponding to two edge nodes e1 and e2
respectively, from Equation (4.22).
Condensing an original tangent matrix induces important effects which are intro-
duced and illustrated in the following sections.
4.9.2 Nonzero Fill-ins in Sparse Tangent Matrix for AH-Adaptive Mesh
The current research on the AH-adaptive analysis utilizes the IBM WSMP solver
[30], a sparse matrix solver which only utilizes information from the nonzero components
and saves it into linear arrays in order to reduce computational overhead. The nonzero
enforcement effect, which will be illustrated in this section, for the condensed matrix is
essential because it is necessary to determine the exact location of all nonzero positions
within the matrix in order to properly use the solver.
The condensation method ([28]) actually can be perceived as splitting the columns
and rows of the constrained DOFs into the columns and rows of the DOFs on which
102
they are dependent. To illustrate the perception, an example for the expanded number
of DOFs per node in a mechanical analysis is demonstrated in Figures 4.5, 4.6 and 4.7.
1. On the basis of the mesh in Figure 4.5, the DOFs on nodes 2 and 11 need to be
constrained. Due to the expanded number of DOFs in a 3-D mechanical analysis,
and the boundary conditions (prescribed displacements), a reference of each DOF
and the equation number is included in the figure.
2. The original nonzero components in the tangent matrix are presented in Figure
4.6. Splitting the columns and rows of the constrained DOFs in the matrix induces
nonzero enforcement effects into some positions which are originally zeros.
3. Figure 4.7 shows the condensed tangent matrix, and all the nonzero components
including those caused by the nonzero enforcement effect.
Note that because this example mesh contains only three elements and 16 nodes,
the non-zeros appear very dense in the tangent matrix (Figures 4.6 and 4.7). For practical
structures, they contain more elements and DOFs so that the non-zeros will be sparse.
4.9.3 Residual Array
The similarity and difference in condensing the residual array and tangent matrix
is illustrated in [35].
4.10 Gauss Point Quantities Balancing
New Gauss points, which replace all old Gauss points, will always be generated
during either element refinement or coarsening. However, 1) coarsening does not create
103
1
2
12
7
3
4
5
6
8
9
10
11
13
14
15
16
Node DOF Equation
1 x
y
zx
y
z
x
y
z
x
y
z
x
y
zx
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
z
x
y
zx
y
z
x
y
z
x
y
z
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
2
3
4
5
67
8
9
10
11
1213
14
1516
17
1819
20
2122
23
24
25
26
2728
29
3031
32
3334
35
36
3738
39
40
41x
yz
Fig. 4.5. Splitting the row and column of the dependent DOF.
104
: nonzero elements X
Original tangent matrix for the system of equations
Fig. 4.6. Nonzero fill-in effect induced by condensing the tangent matrix.
105
: original nonzeros in the stiffness matrix
: nonzero enforcements (fill-ins) due to the condensation effect
X
Condensed tangent matrix
Equation
Equatio
n
number i
n
number i
n
origin
al matri
x
origin
al mat
rixNew
equation n
umber
New equat
ion n
umber
Fig. 4.7. Nonzero fill-ins in the condensed matrix.
106
new nodes at all, and 2) refining does not necessarily involve with generating new nodes
(Sections 4.6 and 4.7). The nodal solution magnitudes with which to start for the next
time increment may be inconsistent with the Gauss point magnitudes.
Figure 4.8 shows an example of the possible inconsistency:
1. Figure 4.8(a) contains elements which are to be re-meshed. At this time increment,
one element is going to be refined to form a new adaptive mesh in Figure 4.8(b).
New Gauss points for the element are generated with data transferred. Meanwhile
all DOFS become free (unconstrained).
2. At a later time increment, two elements in Figure 4.8(b) are to be coarsened. Note
that new Gauss points of the coarsened elements are processed at the re-meshing
stage. And there are no new nodes created at all, all nodal solutions inherit from
those in the previous time increment.
3. However, after the new adaptive mesh is formed and upon identifying constrained
nodes, the mesh (Figure 4.8(c)) has two constrained nodes. The nodal solutions
on these constrained nodes now are forced to satisfy their individual constraint
equations, regardless of how much the nodal magnitudes have to be adjusted.
Therefore, on the elements which were just coarsened, the distributions of Gauss
point quantities (such as stresses, strains) become inconsistent with nodal quantity (dis-
placement) magnitudes. This inconsistency:
• does not occur in a thermal analysis, as there are no Gauss point quantities to
transfer during re-meshing. And the calculation of heat source loads for each
107
element on Gauss points is performed after the adaptive mesh information (such
as identifying constrained nodes) has been processed.
• may deteriorate convergence, especially in mechanical analyses of elasto-plasticity
or applying large deformation theory, which already have slower convergence rates
or are easier to diverge originally.
• may also give incorrect solutions, if solving the incremental equation system start-
ing with an incorrect combination of nodal and Gauss point quantity magnitudes.
To combat this, after a new adaptive mesh is determined and all constraint equa-
tions are acquired, the magnitudes of Gauss point quantities are adjusted. By using the
displacement results, and temperature inputs if a thermal-mechanical analysis such as
welding simulations, which 1) are from the previous time increment, and 2) satisfying
the constraint equations of the new adaptive mesh, the system of equations is solved
with one iteration starting with the interpolated Gauss point quantities. This yields a
new magnitude distribution for the Gauss points, balanced with the nodal quantities.
4.11 Recovering the Condensed DOFs
After the condensed system has been solved, the entire solutions for the origi-
nal/complete system can be acquired by applying the constraint equations to calculate
the magnitudes of the constrained DOFs.
108
(a)
(b)
(c)
: elements which are to be / have been refined
: elements which are to be / have been coarsened
: elements which remain unchanged
: new constrained nodes
Fig. 4.8. Balancing between nodal and Gauss point quantities
109
4.12 Convergence Efficiency Improving Strategy
Among the causes which make a mechanical analysis slower to converge are such
as plasticity, and large deformation simulation. This tendency may also induce diver-
gence at times during the entire simulation. (The algebraic equation system solving is
considered to be divergent also if not converging after a certain number of iterations.) In
ordinary finite element analyses, a time step is cut back upon diverging. Adaptive mesh
analysis process new mesh information (re-meshing, identifying constraint equations,
non-zero fill-ins determinations, etc.) each time a new mesh is generated. Therefore, a
cutback means spending more CPU usage on re-doing the mesh. As shown in Figure
4.1 of block group for “convergence efficiency improving”, to save the computational
cost, the AH-adaptive analysis scheme does not generate a new adaptive mesh if there
is divergence occurred. Instead, the scheme utilizes this adaptive mesh for the cutback
time step. The element density will be adequate for the cutback as long as the sizes of
moving forced refinement (Section 2.5) are large enough to include a tolerance for the
need of a few consecutive time steps.
Having constrained DOFs tends to make the equation system slower to converge.
This adds to the occurrence possibility that utilizing the same mesh for the cutback(s)
may not prevent the system from diverging. Therefore, to help a mechanical analysis
excess the occasional convergence hurdle, a maximum number of allowed cutbacks with
the same mesh in set-up for an AH-adaptive mechanical analysis. If the system still
diverges after the allowed number of cutbacks, the AH-adaptive scheme recalls a pre-
viously convergent mesh (node coordinates, solutions, connectivity, etc.) for this time
110
increment. Through this, the system is not locked in divergence and can step on until
the entire simulation is finished.
Thus, the AH-adaptive scheme also saves the information of the latest adaptive
mesh which has the system convergent, when this time increment is finished. This mesh
may be recalled if needed later on.
111
Chapter 5
Mechanical Analyses Numerical Examples
5.1 Linear Weld Path — Comparison of Static and AH-Adaptive Anal-
ysis
In order to 1) evaluate the computational cost using both the static and the
AH-adaptive schemes, and 2) verify that the adaptive analysis results match the static
analysis results, comparisons between the two analyses are performed.
The structural analysis is based on the temperature results from the same model
in [35] as inputs. Figure 5.1 depicts both the original mesh, and the boundary conditions
(prescribed displacements):
• At the front end, the node at the bottom of the mid point along y-direction is fixed
with δy = δz = 0.
• At the opposite end, the node at the bottom of the mid point is fixed at all DOFs.
And a neighboring node is fixed with δz = 0.
Material properties for ASTM 131 grade EH-36 steel are used [32]. Eight-node
hexagonal brick-type elements (hex8) are utilized in the analyses.
112
5.1.1 Hardware and Software
The simulations were performed on a SGI altix 350 system with 8 CPUs. The
software used in this study is an in-house finite element code written in Fortran 90.
An implicit solution scheme and the Newton-Raphson method were used to solve the
non-linear problems in an iterative fashion.
5.1.2 Analysis Results
Figure 5.2 shows the structural response simulation of the static mesh (Figure
5.1) analysis, using the temperature results presented in the previous paper [35]. In the
figure, the blue shaded elements are the deformed shape at t = 3600 sec, in reference
with the undeformed shape. An adaptive mesh analysis is also performed. Figure 5.3
depicts the initial coarse mesh for the structural analysis. The deformation results at t
= 3600 sec is shown in Figure 5.4, with red shaded elements of the deformed in reference
with the undeformed shape.
5.1.3 Comparison
Table 5.1 gives the computation statistics for both the static and the AH-adaptive
analyses. Note that the presented examples demonstrate the ability to reduce the analysis
CPU time by 1131.25 % compared to the conventional static solution.
113
X
Y
Z
23
X
Y
Z
Opposite end
Front end
Fig. 5.1. Static mesh with boundary conditions
114
X
Y
Z
1.71+00
default_Deformation :Max 1.71+00 @Nd 122
X
Y
Z
Fig. 5.2. The deformation results (mm) of the static mesh analysis. Magnificationfactor = 5.0.
115
X
Y
Z
X
Y
Z
Fig. 5.3. Initial mesh for the AH-adaptive analysis
116
X
Y
Z
default_Deformation :Max 1.25+00 @ Nd 25
X
Y
Z
Fig. 5.4. The deformation results (mm) of the adaptive mesh analysis. Magnificationfactor = 5.0.
117
Table 5.1. Comparison between the static and the AH-adaptive analyses on the modelStatic Mesh Analysis AH-Adaptive Analysis
Number of Nodes Statically 77979 Initial 240 Peak 28030Number of Elements Statically 70800 Initial 120 Peak 19174Total Number ofTime Increments
830 915
Total number if Iter-ations
5322 6578
Maximum displace-ment (mm)
1.71 1.25
Total Analysis CPUTime
105682 sec 72631 sec
Processing adaptivemesh information
(N/A) 52008 sec
Residual and tan-gent matrix assem-bling
10935 sec 4507 sec
Algebraic equationsolving
87310 sec 13478 sec
118
5.2 Combined Weld Path (Curved and Linear) — Evaluation of AH-
Adaptive Analysis Scheme
The AH-adaptive FE scheme is also applied to simulate a welding procedure on a
3ft× 3ft plate shown in Figure 5.5. Note also the heat source does not merely move in
linear paths, but also moves along a one-quarter arc of a circular path. In this case, the
heat source is a hybrid of laser-GMAW weld, with heat input parameters can be seen in
[35]. The material properties, hardware and software are the same as those described in
Section 5.1.
The performed structural response simulations are based on the temperature re-
sults acquired from the same structure of [35] — the example in Section “Combined
Weld Path (Curved and Linear)”. The deformation results are shown in the following
section.
5.2.1 Analysis Results
The experimental result of deformations is shown in Figure 5.6. The plate buckles
after applying the heat source. And the dominant buckling in this specific experimental
case is the third mode (the first three buckling modes depicted in Figure 5.7). Simulation
results at t = 3600 sec applying Total Lagrangian formulation for large deformation
analyses with different parameter settings are shown in Figures 5.8 and 5.9. Figure 5.8
uses the permissible gradient = 58, while Figure 5.9 sets to 400. Both analyses allow
elements to be refined to minima of 0.3mm along thickness direction, and 1.0mm for
the remaining two directions. As the figures show, buckling does also occur in both
119
simulations. Figure 5.8 captures the first buckling mode. Meanwhile the buckling mode
on Figure 5.9 is the second mode.
1. In experiments, imperfections of structure shape can induce different deformation
magnitudes, especially when buckling occurs.
2. In simulations of adaptive analyses, different mesh densities controlled by user
settings may also result in different buckling modes between simulations, because
of the different connectivities and therefore the allowed deformation shape through
the entire analyses.
3. Experimental boundary conditions usually will not be exactly identical to those
applied in a simulation. Thus, if the structure buckles, the maximum buckling
displacement magnitudes may differ much between experiment and simulation, or
even between simulations especially if they capture different buckling modes.
Solving this problem using a conventional static mesh is intractable due to:
• the large number of elements and DOFs which would be required to adequately
describe the structure.
To generate a static mesh for the 3ft×3ft plate (of the same size shown in Figure
5.5), the estimated node number is calculated from the static mesh of Figure 5.1
used in Section 5.1. The small structure in Figure 5.1 is cut from the 3 × 3 plate,
and contains only 152.4mm of the linear weld path portion. Its width is also only
304.8mm (less than the width 914.4mm of the entire plate). The total weld length
on the 3 × 3 plate is 609.6mm (two linear segments) + 215.8mm (circular path)
120
X
Y
Z X
Y
ZX
Y
Z X
Y
Z
Fixed at the bottom, x=y=z=0
Fixed at the bottom, x=0Fixed at the bottom, z=0
Fixed at the bottom,y=0
Fig. 5.5. The plate and the initial mesh.
121
Fig. 5.6. Experimental buckling results.
122
Mode 3
Mode 1Mode 2
1
2
3
Pure Angular Distortion
Fig. 5.7. 1st – 3rd buckling modes and pure angular distortion.
123
X
Y
Z
8.49+00
default_Deformation :Max 8.49+00 @Nd 25
X
Y
Z
Fig. 5.8. The deformation result (mm) at t = 3600 sec, with permissible gradient (peaktemperature) = 58 ◦ C. (Magnification factor = 2.5)
124
X
Y
Z
1.35+01
default_Deformation :Max 1.35+01 @Nd 21
X
Y
Z
Fig. 5.9. The deformation result (mm) at t = 3600 sec, with permissible gradient (peaktemperature) = 400 ◦ C. (Magnification factor = 2.5)
125
= 825.4mm. Therefore, considering there are 77979 nodes on the mesh of Figure
5.1, using similar mesh densities for the 3ft × 3ft plate would give the estimated
node number to be “above” 77979/152.4 ∗ 825.4 = 422335.
• the complexity of making transition elements (between the high element density
areas and elsewhere), especially because of the existence of the circular weld path.
The CPU usages and other statistics using different sets of parameters (such as
the sizes of the moving forced refinements) with the AH-adaptive scheme is presented in
Table 5.2. Using a standard direct sparse solver, the computational cost to factorize and
solve the equation system is known to grow as O(n2) for 3D simulations (and O(n32 ) for
2D simulations) [34], where n is the total number of equations in the system. However,
benefitting from the huge reduction of DOFs, in the AH-adaptive analysis the equation
solving is not dominating the total computation. And the adaptivity overhead induced
by processing the adaptive mesh information is designed to be linear (O(n)). If the
necessary DOFs for a static mesh analysis is larger, the efficiency is expected to be
even much higher because the computational expense spent on decomposing the tangent
matrices will be reduced more significantly.
126
Table 5.2. Statistics of the AH-adaptive analysisCase number L1 L2Radii of EnforcedRefinement
11.987mm and 13.91mm 11.987mm and 13.91mm
Minimum Length ofElements
0.3mm in thickness 0.3mm in thickness
0.6mm elsewhere 0.6mm elsewherePermissible Gradient(Peak Temperature ◦C)
400.0 58.0
Total Number ofTime Increments
480 507
Number of Nodes Initial 440 Peak 22151 Initial 440 Peak 24084Number of Elements Initial 288 Peak 17983 Initial 288 Peak 16398Average number ofnodes per iteration
14152 15008
Total Analysis CPUTime
75107 sec 85731 sec
Residual and Tan-gent Matrix Assem-bling
50031 sec 57532 sec
Algebraic EquationSolving
14048 sec 16130 sec
Adaptive Mesh In-formation Processing
10168 sec 11094 sec
127
Chapter 6
Comparisons between using Condensation Theory,
Lagrange Multiplier and Penalty Method
for Constrained DOFs
Upon applying h-refinement strategies [20] in finite element simulations, depen-
dent (also referred to as “constrained”) nodes occur wherever adjacent elements have
different element generations. Setting the element generations to be identical (for exam-
ple, 0) for all elements on the starting mesh in Figure 2.11(A), Figure 2.11(B) shows an
example mesh containing constrained nodes due to the generation differences. The de-
pendent nodes need to satisfy the interpolation (linear for hex8 elements, and quadratic
for hex20 elements) on the interface shared with the adjacent elements. Thus, the DOFs
on these nodes must be constrained, and the original system of equations therefore has to
be adjusted before being solved to satisfy these constraints. A few methods are available
to account for the constrained DOFs:
• Condensation and recovery theory [28]
• Lagrange multiplier [38]
• Penalty Method [38]
The examples given in this paper use hex8 elements whenever a specific mesh is
needed. However, the utilization of these methods for dealing with constrained equations
apply to all element types.
128
6.1 Objective
The objective of this chapter is to compare the above methods which serve to
process an algebraic system of constrained equations. The comparisons are effective
not only when applying the AH-adaptive scheme [35, 39] for simulations that involve
multi-scale analyses, but also for other equation systems induced by h-refinement.
6.2 Determination of Constraint Equations
In Figure 2.12 consisting of hex8 elements, nodes 1–10 are constrained by the
corner nodes A–D on the surface of element a, because only linear interpolations will
be allowed on that surface. By using the corresponding iso-parametric coordinates (r1
and r2), every constrained node may be treated as depending on the four corner nodes
of the surface. Hence the constraint equation for a dependent node is:
Td
=(1 − r1)(1 − r2)
4TA
+(1 + r1)(1 − r2)
4TB
+(1 + r1)(1 + r2)
4TC
+(1 − r1)(1 + r2)
4TD
(6.1)
where r1 and r2 are the iso-parametric coordinates of the constrained node on the surface.
Td
is the temperature of the dependent node. TA
, TB
, TC
, TD
are the temperatures on
the four corner nodes forming the element surface.
For a dependent node located on an edge of an interface, Equation (6.1) will reduce
to two terms only, which correspond to the fact that this node is actually constrained by
the two end points of the edge. For nodes on these edge interfaces, the equations are:
129
Td
= αTe1 + βT
e2 (6.2)
where node d depends on two edge nodes e1 and e2, α and β are the coefficients corre-
sponding to the two edge nodes e1 and e2, respectively, from Equation (6.1).
For instance, in Figure 2.12 nodes 1 and 2 only depend on nodes A and B, while
nodes C and D do not affect them at all. Therefore,
T1 = 0.5 ∗ TA
+ 0.5 ∗ TB
(+0 ∗ TC
+ 0 ∗ TD
)
T2 = 0.25 ∗ TA
+ 0.75 ∗ TB
(+0 ∗ TC
+ 0 ∗ TD
)
6.3 Condensation and Recovery Theory
To compute a system containing some constraint equation(s), the condensation
(and recovery) theory takes the constrained variables (DOFs) out, in order to form a
equation system that has only free variables (DOFs). The condensed variables are recov-
ered by the constraint equations after the solutions of the free variables are calculated.
6.3.1 System Condensing
For a non-linear system, the incremental nodal solution is computed from the
algebraic system
Aδu = b (6.3)
130
where A is the tangent matrix with b being the negative of the residual, and δu is the
incremental solution during an iteration.
For each iteration, the system (Equation (6.3)) is processed by partitioning the
DOFs into
{δu} =
δur
δuc
(6.4)
where the subscript r stands for “retained”, and c stands for “condensed (out)”. Thus,
δur represents the actual DOFs to be retained, and δuc represents the condensed DOFs
of the dependent nodes. Thus, the entire partitioned non-linear system can be repre-
sented as:
Arr Arc
Acr Acc
δur
δuc
=
br
bc
(6.5)
The general representation of the constraint equations is given by
[Cr Cc
]
δur
δuc
={
Q
}(6.6)
where Cr
and Cc
are the coefficients for the retained and condensed nodes, respectively,
131
and Q is the constant in the system of constrained equations. For the AH-adaptive
analysis scheme, a constraint equation must have the form of
uc
=N∑
k=1C
kuk
(6.7)
where uc
is the solution of the constrained node. N is the number of nodes on which it
depends, and Ck
is the corresponding constraint coefficient.
The RHS terms in Equation (6.7) can be moved to the LHS of the equation:
uc−
N∑k=1
Ckuk
= 0 (6.8)
So that the constant term Q in Equation (6.6) must be zero.
Utilizing the equations representing the constraints, we now have:
[Cr Cc
]
δur
δuc
={
0
}(6.9)
So,
{δuc
}= −[ Cc
]−1[ Cr]{
δur
}= [ Crc
]{
δur
}(6.10)
where Crc = −C−1
cCr is the combined coefficient matrix.
132
By substituting Equation (6.10) into Equation (6.5),
[Arr + ArcCrc + CT
rcAcr + CT
rcAccCrc
] {δur
}=
{br + CT
rcbc
}(6.11)
the solutions for the “retained” DOFs{
δur
}is:
{δur
}=
[Arr + ArcCrc + CT
rcAcr + CT
rcAccCrc
]−1 {br + CT
rcbc
}(6.12)
And the “condensed” DOFs can then be recovered by Equation (6.10).
6.3.2 Example
In Figure 2.11, with 24 independent nodes and 4 constrained nodes, the constraint
equations for the dependent nodes are:
ua
=12(u1 + u2) (6.13)
ub
=12(u2 + u3) (6.14)
133
uc
=12(u4 + u5) (6.15)
ud
=12(u5 + u6) (6.16)
Equation (6.9) for this example system becomes
1 0 0 0 −12 −1
2 0 0 0 0 0 ... 0
0 1 0 0 0 −12 −1
2 0 0 0 0 ... 0
0 0 1 0 0 0 0 −12 −1
2 0 0 ... 0
0 0 0 1 0 0 0 0 −12 −1
2 0 ... 0
4∗28
ua
ub
uc
ud
u1
u2
u3
u4
u5
u6
u7
...
u24
28∗1
=
0
0
0
0
4∗1
(6.17)
134
Thus,
[Cc
]=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
= [ I ]4∗4 (6.18)
[Crc
]= −[ Cr
] =
12
12 0 0 0 0 0 ... 0
0 12
12 0 0 0 0 ... 0
0 0 0 12
12 0 0 ... 0
0 0 0 0 12
12 0 ... 0
4∗24
(6.19)
Therefore, the retained, and the condensed, DOFs can be solved using Equation
(6.12).
6.3.3 Restrictions
To utilize the condensation (and recovery) theory in AH-adaptive analyses, the
followings need to be taken care of:
135
1. the tangent matrix is condensed to a smaller size [28, 35], so allocation for the
matrix of the corresponding dimension is necessary.
2. in practical simulations, the tangent matrix is generally a sparse matrix so that
a linear sparse matrix solver can be used to enhance computational efficiency.
However, condensing the tangent matrix induces the effect of nonzero fill-in which
is illustrated in Tsau et al. [35]. Special considerations have to be performed to
accurately solve the system.
6.4 Lagrange Multiplier
Lagrange multiplier method is a widely used procedure for imposing constraints
onto a system. The method adds the constraints, and operates on, the original variational
formulation of the system, in order to acquire the adjusted equations to solve.
6.4.1 Equation Derivation
Consider the variational formulation of a discrete structural model,
Π =12uTAu − uTb (6.20)
and
∂Π∂u
i
= 0 for all i (6.21)
136
where Π is the total potential (also referred to as “functional”) of the system, u is the
(incremental) solutions of all DOFs, ui
is the (incremental) solution of the i − th DOF,
A is the tangent matrix, and b is the residual.
Assume that there are m linearly independent constraints Cu = Q to be imposed,
where C is a coefficient matrix of order m×n, n is the total number of equations in the
original system, Q is an m × 1 array of constants. By applying the Lagrange multiplier
method, the variational formulation of a discrete structural model [40] is modified as
Π∗(u, λ) =12uTAu − uTb + λ
T(Cu − Q) (6.22)
where λ is a vector of m Lagrange multipliers.
Letting δΠ∗ = 0, and because δu and δλ are arbitrary,
A CT
C 0
u
λ
=
b
Q
(6.23)
As illustrated in Equations (6.7) and 6.8 (or Equation (6.17) from a numerical
example), in the constraint equations it must be Q = 0.
And,
A CT
C 0
u
λ
=
b
0
(6.24)
Multiplying −CA−1 with the first row, and adding to the second row,
137
A CT
0 −CA−1CT
u
λ
=
b
−CA−1b
(6.25)
The vector λ is acquired first by
λ = (−CA−1CT)−1(−CA−1b) (6.26)
This allows to determine u:
u = A−1(b − CTλ) (6.27)
6.4.2 Restrictions
The Lagrange multiplier method induces many drawbacks:
1. though the construct of A is generally a sparse matrix, −CA−1CT in Equation
(6.26) is a dense matrix, so that it can not be operated by a sparse matrix solver.
Figures 6.1 shows a mathematical example of how −CA−1CT forms a dense/full
matrix, even if A−1 is set to be sparse.
Note that for using a standard direct sparse solver, the computational cost to fac-
torize and solve the equation system is known to grow as O(n2) for 3D simulations
[34], where n is the total number of equations in the system. However, the opera-
tions on the dense matrix −CA−1CT is of order O(m3), where m is the number
of constrained equations.
138
2. as Equations (6.26) and (6.27) show, the Lagrange multiplier method requires
computations of more matrix inverses (in addition to other arithmatic operations)
which are the major contributing factor to computational cost especially for large
structures,
3. the tangent (stiffness) matrix A also retains the total number of DOFs, and does
not benefit from the reduction in dimension realized because of the constrained
DOFs in applying condensation theory.
6.5 Penalty Method
Similar to the Lagrange multiplier method, penalty method combines the imposed
constraints to the original variational formulation of the system. However, instead of the
additional variable λ, the penalty method introduces a constant penalty number α to
derive for an modified equation system to solve.
6.5.1 Utilizing Penalty Number in a System
Starting also from Equations (6.20) and (6.21), the variational formulation of a
discrete structural model, and the assumption that there are m linearly independent
constraints Cu = Q to impose. In the penalty method, the variational formulation of a
discrete structural model [40] is formulated as follows:
Π∗∗(u) =12uTAu − uTb +
α
2(Cu − Q)T(Cu − Q) (6.28)
where α is a constant large number, and α max(Aij
).
139
1 2 3 4 5 6 7 8 9
1 X X
2 X X X
3 X X X
4 X X X
5 X X X
6 X X X
7 X X X
8 X X X
9 X X
U1 = 0.5 * (U7+U8)U2 = 0.5 * (U8+U9)U3 = 0.5 * (U6+U7)U4 = 0.5 * (U7+U8)
1 2 3 4
1 X
2 X
3 X
4 X
5
6 X
7 X X X
8 X X X
9 X
1 2 3 4 5 6 7 8 9
1 X X X
2 X X X
3 X X X
4 X X X
Constraint Equations
A-1
CT
C ( (X : Nonzero component
1 2 3 4 5 6 7 8 9
1 X X X
2 X X X
3 X X X
4 X X X
A-1C
TC
1 2 3 4
1 X X
2 X X X
3 X X X
4 X X
5 X X
6 X X X
7 X X X X
8 X X X X
9 X X X
1 2 3 4
1 X X X X
2 X X X X
3 X X X X
4 X X X X
A-1C
TC
Fig. 6.1. The nonzero components in C, A−1, CT , and the resulted matrices.
140
Letting δΠ∗∗ = 0, and considering that in AH-adaptive analyses, the Q must be
a constant matrix of zeros,
(A + αCT C)u = b (6.29)
The solution for u is then
u = (A + αCT C)−1b (6.30)
6.5.2 Restrictions
The disadvantages of using the penalty method have been known, and are stated
in Ref [38]:
1. the penalty number is generally problem-defined with a specific formulation,
2. a high penalty number can cause the resulted matrix (A + αCT C) to be ill-
conditioned, because the off-diagonal components are multiplied by such a large
number α.
141
Chapter 7
Conclusions
An Anisotropic h-Adaptive (AH-adaptive) analysis scheme has been developed
using condensation and recovery methods. The AH-adaptive procedure can reduce the
numbers of DOFs significantly compared to static or even isotropic adaptive analyses.
This results in significant reductions in CPU usage requirements, since the number of
DOFs in the resulting tangent matrices is an important part in computational cost (as
the number of nodes increases, the CPU time spent dealing with tangent matrices can
dominate the entire analyses).
The proposed mesh refinement procedure is based on the previous solution dis-
tribution. The procedure can also use, if available, knowledge of areas of expected high
gradients, so computationally expensive iterative examination of the mesh convergence is
minimized. The AH-adaptive FEA method enhances computational efficiency especially
for multi-scale analyses. Not only can the scheme be applied to welding or laser forming
analyses, but it may also be used for other areas that seek to employ large simulation
models subjected to disturbances that result in steep field gradients confined to relatively
small localized areas.
Using static meshes on multi-scale/large structure analyses is very labor-consuming,
especially because of meshing the elements in transition regions. On the other hand, ap-
plying the AH-adaptive scheme only needs a very coarse initial mesh. All transition
142
elements, and the necessary element densities to sufficiently integrate the heat source,
are generated automatically by the control criteria – the gradient measures and the
moving forced refinements.
Comparisons between the static mesh analyses in Ref. [33] and the AH-adaptive
mesh analyses on the 3ft × 3ft plate in this work demonstrate that the AH-adaptive
analysis simulation provides a 99.63% reduction of CPU usage on the same hardware,
and a reduction of 93.805% compared to the wall-clock time spent for the parallelized
computing simulation on a 16-CPU machine.
If buckling occurs, the captured mode and the maximum displacements may not
be the same between finite element simulations and experiments. This is because of
structure imperfections (in experiments), element densities, and boundary conditions,
etc.
In an AH-adaptive mechanical analysis, in addition to nodal peak temperatures if
there are thermal results as the input, a few quantities such as plastic strains or stresses
may serve as a basis for the gradient measures. The quantities may also be combined to
evaluate gradients for elements.
Utilizing the Lagrange multiplier method in AH-adaptive analyses for constraint
DOFs deteriorates the numerical efficiency due to its demand for more matrix operations,
primarily computationally expensive matrix inversions. Furthermore, some steps may
require operating on a full matrix form which is computationally more expensive than a
sparse solver.
Though the penalty method avoids such additional costly operations, determining
the penalty number is a critical but difficult task. And the ill-conditioning caused by a
143
too high penalty number also reduces the solvability of the algebraic equation systems.
Despite the fact that using the condensation theory needs to deal with the nonzero fill-
in effects due to condensing the tangent matrix, it provides a better combination of
computational efficiency and stable solvability. Therefore, the theory is preferred for
processing the constraint DOFs in the AH-adaptive finite element analysis scheme.
144
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Vita
Shih-Horng Tsau received his B.S. degree in Power Mechanical Engineering at
Tsing-Hua University in Taiwan in June 1993, and received his M.S. degree in Mechanical
Engineering at National Cheng-Kung University in Taiwan in June 2001. In Aug 2001,
he enrolled in the graduate program in Mechanical Engineering at Pennsylvania State
University and began to pursue his Ph. D. degree. His research interests include solid
mechanics, thermal processing, nonlinear finite element analysis, numeric methods, as
well as adaptive mesh analysis algorithm and coding.