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MATH 2034 Lecture Notes 10-2 Dr. Yan DING Dept. of Maths & Stats. RMIT University

Solution of Non-linear FE Models

The cause of nonlinearity:

The FE solution to a time-independent problem always

involves of solving a set of simultaneous algebraicequations of the following form:

In linear analysis, both [K] and {F} are regarded asindependent of {a}.

Whereas in nonlinear analysis, [K] and/or {F} are regardedas functions of {a}.

The followings are two examples of nonlinearity.

{ } { }FaK =][

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Material nonlinearity:

Stiffness matrix is composed of a constant term [Ko]and a term [KN] that depends on deformation. Thus:

([Ko]+[KN]){a}={F}

where: [KN] = f ({a}), i.e. depends on deformationof the structure.

Geometric nonlinearity:

Consider the plane cantilever beam, we seek the

quasistatic deflection produced by loads P and ML.

Assuming that the beam is slender and that its

material is linearly elastic at all times. For small

deflections, linear theory is adequate, and the rootmoment is: Mo =P LT + ML.

For large deflections, the moment arm H of force P

is less that LT, thus:

Mo =P H + MLwhere H depends on P and ML.

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In the solution of non-linear problems, we will always obtain a set

of algebraic equations:

(a) = F - P(a)where a is the set of discretization parameters, F is a vector that isindependent of the parameters, and P is a vector dependent on theparameters, a. These equations may have multiple solutions, asshown below. Thus, if a solution is achieved it may not necessarilythe solution sought. In order to obtain realistic answers, small-step

increment approaches from known solutions are essential.

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The general nonlinear problem should always be formulated as the solution of:

n+1= (a n+1) = F n+1 - P(a n+1) = 0 (1)which stars from a nearby solution at:

a = an, n= 0, F= Fn (2)and often arises from changes in forcing function Fn to

Fn+1 = Fn + Fnthe determination of the change such that

an+1 = an + anwill be the objective and generally the increments ofFn will be kept reasonablysmall in order to follow the path dependence.

The solution of the problem above can not be solved directly and will alwaysrequire some sort of iteration (repeated solution of linear equations). There aremany iterative techniques for solving non-linear problems. Among them, the

Newton-Raphson method is the most rapidly convergent process for solutions ofnon-linear problems.

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Solutions of Nonlinear FE Models

The Newton-Raphson Method

In the Newton-Raphson iterative method, to the first order, Eq(1)

can be approximated as:

(3)

Here the superscript i indicates the iteration number and usually

starts by assuming

a1n+1 = an

in which an is a converged solution at a previous load level or time

step. The Jacobian matrix corresponding to a tangent direction is

given by

Eq(3) gives immediately the iterative correction as

0)()(1

1

1

1 =

+ +++

+

i

n

i

n

i

n

i

n daaaa

aaPKT ==

i

n

i

T

i

n

i

n

i

n

i

T KdaordaK 11

1 )( +

+ ==

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Solutions of Nonlinear FE Models

The Newton-Raphson Method

A series of successive approximations gives

where

The process illustrated in the figure below shows the vary rapidconvergence:

i

nn

i

n

i

n

i

n

aa

daaa

+=

+= ++++ 111

1

=

=i

k

k

n

i

n daa1

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Thermal & Thermal-Stress Analyses

The following procedures are discussed:

How to do a thermal analysis;

How to apply thermal loads in a stress analysis;

How to do a coupled-field analysis.

These will be done through the following two

sections:

1. Thermal analysis

2. Thermal-stress analysis

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1. Thermal Analysis

Thermal analyses are used to determine the

temperature distribution, thermal gradient, heat flow,

and other such thermal quantities in a structure.

A thermal analysis can be steady-state or transient:

Steady-state implies that the loading conditions have settled

down to a steady level, with little or no time dependency.

Transient implies that conditions are changing with time. A

typical example is a casting in the process of cooling down

from molten metal to solid.

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1. Thermal Analysis

Thermal Loading Conditions

Perfectly insulated surfaces where no heat transfer

takes place.

Adiabatic surfaces:

Surfaces where heat transfer occurs by means of

radiation. Input consists of emissivity, Stefen-Boltzmann

constant, and optionally, temperature at a space node.

Radiation:

Regions where the volumetric heat generation rate is

known

Heat generation:

Points where the heat flow rate is known.Heat flow:

Surfaces where the heat flow rate per unit area is known.Heat flux:

Surfaces where heat is transferred to (or from)

surroundings by means of convection. Input consists of

film coefficient h and bulk temperature of thesurrounding fluid Tb.

Convections:

Regions of the model where temperatures are known.Temperatures:

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1. Thermal Analysis

Element Attributes

1) Thermal element types:

Thermal analyses use thermal elements only. A thermal element

has only one DOF per node.

The commonly used thermal element types are:

2-D Solid 3-D Solid 3-D Shell Line Elements

Linear PLANE55 SOLID70 SHELL57 LINK31, 32, 33, 34

Quadratic PLANE77

PLANE35

SOLID90

SOLID87

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1. Thermal Analysis

Element Attributes

2) Material Properties:

Minimum requirement is the thermal conductivity, KXX.

Specific heat (C) is required if internal heat generation is to beapplied.

ANSYS supplied material library contains both structural and

thermal properties for a few materials. Generally the analystscreate and use their own material library.

3) Real Constant:

Primarily needed for shell and line elements

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1. Thermal Analysis

Thermal Loading

Prescribed Temperatures:

DOF constraints for a thermal analysis:

Solution > -Load- Apply > Temperature

Convections: These are surface loads:

Solution > -Load- Apply > Convection

Adiabatic surfaces:

Perfectly insulated surfaces where no heat transfer takes place.

This is the default condition, i.e, any surface with no boundaryconditions specified is automatically treated as an adiabaticsurface.

Other possible thermal loads:

Heat flux (BTU/Hr-in2 or W/m2)

Heat flow (BTU/Hr or W)

Heat generation (BTU/Hr-in3 or W/m3)

Radiation (BTU/Hr or W)

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1. Thermal Analysis

Results

The results of a thermal analysis are written to a result file, jobname.rth, as

well as to the in-memory database.

Review results typically consists of contour plots of temperature, thermalgradient, and thermal flux:

General Postproc > Plot Results > Nodal Solu (or Element Solu )

A useful option for contour plots in 3-D solid models is isosurfaces, whichare the surfaces of a constant value:

Utility Menu > PlotCtrls > Style > Contours > Contours Style

Results validation.

Are temperatures within the expected range?

You can generally guess the temperature range based on the prescribedtemperatures and convection boundaries.

Is the mesh adequate?

In the case of stresses, you can plot the un-averaged thermal gradients(element solution) and look for elements with high gradients. These regionsare the candidates for mesh refinement.

If there is a significant difference between the nodal (averaged) and element(un-averaged) thermal gradients, the mesh may be too corse.

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2. Thermal-Stress Analysis

The followings are discussed:

How to apply thermal loads in a stress analysis;

How to do a coupled-field analysis.

These will be done through the following

sections:

A. Overview

B. Sequential Method

C. Direct Method

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Thermal-Stress Analysis

A. Overview

Thermally Induced Stress:

When a structure is heated or cooled, it deforms by expanding

or contracting.

If the deformation is somehow restricted, either by

displacement constraints or an opposing pressure, for example,

thermal-stresses are induced in the structure.

Another cause of thermal stresses is non-uniform deformation,

due to different materials (i.e, different coefficients of thermal

expansion).

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... Overview

There are two methods of solving thermal-stress problems inANSYS. Both methods have their advantages anddisadvantages:

Sequential coupled field: Older method, which uses two element types mapping

thermal results as structural temperature loads.

Efficient when running many thermal transient time points

by few structural time points Can be easily automated with input files

Direct coupled field:

Newer method, which uses one element type to solve bothphysical problems

Allows true coupling between thermal and structuralphenomena.

May carry unnecessary overhead for some analyses.

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B. Sequential Method

The sequential method involves two steps of analyses:

1) First, do the Thermal Analysis

2) Then, do the Structural Analysis

1. The thermal analysis (steady-state or transient) :- Refer to slides No. 10-9 ~ No. 10-14:

Model with thermal elements

Apply thermal loading Solve and review results (the results file: jobname.rth)

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... Sequential Method

2. The static structural analysis:

Switch element types to structural.

Define structural material properties, including thermalexpansion coefficient.

Apply structural loading, including temperatures from

thermal analysis.

Solve and review results (the result file: jobname.rst).

The GUI paths for the above procedure are given in the

next slide.

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... Sequential Method

The Procedure of the Structural Analysis :

a) Move to PREP7 and switch element types from thermal to structural:

Preprocessor > element Type > Switch Elem Type

Select Thermal to Struc, then [OK]Caution: switching element types will reset all element options back to their default settings. For example,

if you used 2D axisymmetric elements in the thermal analysis, you may need to re-specify theaxisymmetric option after the switch. Therefore, make sure to verify and set the proper elementoptions:

Preprocessor > Element Type > Add/Edit/Delete > [Options]

b) Define structural material properties, including the coefficient of thermalexpansion (ALPX).

Caution: If ALPX is not defined or set to zero, no thermal strains will be calculated. By the way, thistechnique can be used to turn off temperature effects, if it is needed !

c) Specify static analysis type. This step is needed only if the thermal analysiswas a transient:

Solution > -Analysis Type- New Analysis

d) Apply structural loads and include temperatures as part of the loading:

Solution >-load- Apply > -Structural- Temperature > From Therm Analye) Solve and review the results.

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C. Direct Method

The direct method involves just one analysis that uses a coupled-

field element type containing all necessary degrees of freedom.

The procedure is:

First prepare the model and mesh using one of the following

coupled field element types:

PLANE13 (plane solid)

SOLID5 (hexahedron) SOLID98 (tetrahedron)

Apply both the structural and thermal loads and constraints to

the model.

Solve and review both thermal and structural results.

Only produces one result file: jobname.rst.

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Sequential vs. Direct

Sequential:

For coupling situations that do notexhibit a high degree of non-linear

interaction, the sequentialmethods is more efficient andflexible because the two analysesare performed independently ofeach other.

Sequential thermal-stress analysisprovides more flexibility inapplying thermal load for a stressanalysis. For example, you canperform a nonlinear transientthermal analysis followed by a

linear static stress analysis, duringwhich the nodal temperaturesfrom ANY load step or time-pointin the thermal analysis can beused as loads for the stress

analysis.

Direct:

Direct coupling is

advantageous when the

coupled-field interaction is

highly nonlinear and is best

solved in a single solution

using a coupled formulation.

Examples of direct coupling

include piezoelectric

analysis, conjugate heat

transfer with fluid flow, and

circuit-electric analysis.

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