FEM Zabaras ComputationalDesign

8/9/2019 FEM Zabaras ComputationalDesign

1/65

CC OO RR NN EE LL LL U N I V E R S I T Y 1

MAE 4700 FE Analysis for Mechanical & Aerospace Design N. Zabaras (11/03/2009)

MAE4700/5700Finite Element Analysis for

Mechanical and Aerospace DesignCornell University, Fall 2009

Nicholas ZabarasMaterials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering101 Rhodes Hall

Cornell UniversityIthaca, NY 14853-3801
http://mpdc.mae.cornell.edu/Courses/MAE4700/MAE4700.htmlhttp://mpdc.mae.cornell.edu/http://mpdc.mae.cornell.edu/http://mpdc.mae.cornell.edu/Courses/MAE4700/MAE4700.html


2/65



Design is the heart of many industries


3/65



References Introduction to optimum design, J. Arora Elements of Structural Optimization, R. Haftka and Z. Gurdal Engineering Design Optimization, lecture notes by N. Olhoff Multidisciplinary System Design Optimization, Lecture notes from

MIT Structural Optimization, Lecture notes from R. Haftka Structural sensitivity analysis and optimization, K. K. Choi and N. H.

Kim Principles of Optimal Design Modeling and Computation, P.

Papalambros and D. Wild Numerical Optimization Techniques for Engineering Design, G.

Vanderplaats Topology Optimization, M. Bendsoe and O. Sigmund
http://books.google.com/books?id=9FbwVe577xwC&dq=introduction+to+optimum+design,+arora,+free-ebook-rapidshare&printsec=frontcover&source=in&hl=en&ei=WPbHSvvoENDL8QbXlajhCA&sa=X&oi=book_result&ct=result&resnum=11#v=onepage&q=&f=falsehttp://books.google.com/books?id=CzgIpexeh7UC&pg=PA3&lpg=PA3&dq=design,+finite+elements,+optimization&source=bl&ots=Q43yxYgoj_&sig=FxBzAanelbkzq1yoGXFoAeAoeZE&hl=en&ei=E1XGSvuqD47WlAeEgs2SAw&sa=X&oi=book_result&ct=result&resnum=4#v=onepage&q=design%2C%20finite%20elements%2C%20optimization&f=falsehttp://www.ime.aau.dk/~no/lecnotes.asphttp://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-888Spring-2004/CourseHome/index.htmhttp://www.mae.ufl.edu/haftka/stropt/http://www.amazon.co.uk/Structural-Sensitivity-Analysis-Optimization-Engineering/dp/038723232Xhttp://www.amazon.co.uk/Structural-Sensitivity-Analysis-Optimization-Engineering/dp/038723232Xhttp://www.amazon.com/exec/obidos/ASIN/0521627273/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0521627273/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0944956017/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0944956017/ref=nosim/mitopencourse-20http://www.amazon.co.uk/exec/obidos/ASIN/3540429921http://www.amazon.co.uk/exec/obidos/ASIN/3540429921http://www.amazon.com/exec/obidos/ASIN/0944956017/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0944956017/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0521627273/ref=nosim/mitopencourse-20http://www.amazon.com/exec/obidos/ASIN/0521627273/ref=nosim/mitopencourse-20http://www.amazon.co.uk/Structural-Sensitivity-Analysis-Optimization-Engineering/dp/038723232Xhttp://www.amazon.co.uk/Structural-Sensitivity-Analysis-Optimization-Engineering/dp/038723232Xhttp://www.mae.ufl.edu/haftka/stropt/http://ocw.mit.edu/OcwWeb/Aeronautics-and-Astronautics/16-888Spring-2004/CourseHome/index.htmhttp://www.ime.aau.dk/~no/lecnotes.asphttp://books.google.com/books?id=CzgIpexeh7UC&pg=PA3&lpg=PA3&dq=design,+finite+elements,+optimization&source=bl&ots=Q43yxYgoj_&sig=FxBzAanelbkzq1yoGXFoAeAoeZE&hl=en&ei=E1XGSvuqD47WlAeEgs2SAw&sa=X&oi=book_result&ct=result&resnum=4#v=onepage&q=design%2C%20finite%20elements%2C%20optimization&f=falsehttp://books.google.com/books?id=9FbwVe577xwC&dq=introduction+to+optimum+design,+arora,+free-ebook-rapidshare&printsec=frontcover&source=in&hl=en&ei=WPbHSvvoENDL8QbXlajhCA&sa=X&oi=book_result&ct=result&resnum=11#v=onepage&q=&f=false


4/65



Analysis vs Synthesis Analysis: Find the properties of a product (predict

consequences) Synthesis/Design: Create a product with desired properties Optimization: Mathematical methods used in the inverse

analysis To design you first need to be able to do analysis Design is much more complex than analysis (non-

uniqueness, non-feasibility, etc.)

Here we are interested in very limited aspects of design:mainly reviewing in some superficial manner optimization &

design of systems analyzed by the FEM i.e. in systemswhere the analysis is implicit, complex and expensive


5/65



Definitions

Design space: all possible designs Objective function : a measure of the quality of the design (user defined)

Constraints: On design requirements, materialavailability, geometry, topology, etc.

Definition of a design problem is not as easy as itlooks

Often conflicting objectives (trade-off)

What is to be considered as an objective and whatas a constraint


6/65CC OO RR NN EE LL LL U N I V E R S I T Y 6


Example

You want to design a bicycle with thefollowing properties: Stiffness Lightness Low cost

Durability Aesthetics

Etc.These are conflicting criteria
http://www.amazon.com/Diamondback-Octane-Mountain-20-Inch-Wheels/dp/B002C8DET2/ref=sr_1_3/189-6321998-3767340?ie=UTF8&s=toys-and-games&qid=1254534282&sr=1-3


7/65



Mathematical statement

A typical design problem can be stated asan optimization problem:

In most designs using FEM all functions(and their derivatives) are defined implicitly

1 2: ( ), , ,...: ( ) , 1, 2,...

o n

i i

inimize g x x xSubject to g G i m

= =

x x

x

Objective function

Constraints

Design space


8/65



Graphical intepretation of an optimization problem


9/65



Sensitivity analysis We need to find the minimum having only a feel of the

slope If we know the value and slope of the functions in the

current point, x (k), then we can approximate the problem

The solution to the above approximate problem shouldprovide a step in the right direction.

Sensitivity analysis is finding the gradients (with respectto the design variables) of the functions of the problem

( ) ( ) ( )1 2

( ) ( ) ( )

: ( ) ( )( ) ..., , ,...

: ( ) ( )( ) ... , 1, 2,...

k k k o o n

k k k i i i

inimize g g x x x

Subject to g g G i m

+ + =

+ + =

x x x x x

x x x x


10/65



Computing sensitivities We dont know the mathematical form of the function we

are dealing with because the problem is implicit. So wecannot just differentiate symbolically.

We can find the gradient by forward finite difference:

This requires one additional direct analysis for each

design variable and thus it can be a time consumingprocess when we have many variables.

( ) ( )( ) ( ) ( )( ) , 1, 2,...,

j

k k i j ik

i j

Sensitivity wrt x

g x gg j n

x

+ = =

x x

x


11/65



The design space is usually large The frame on the right has 11 members. Let us assume

the very simple design problem of findingcross -sectional areas of the truss members.Let us furthermore limit the search to 5different standard cross-section sizes.

The design spaces size is then 5 11 different designs!

You need optimization to search the design space Lets say you want to design the geometry in aturbine to control some properties usually these

properties are implicitly defined in terms of thegeometry using FEM. For example, the temperature,stresses, etc. depend on the shape through a

complex multiphysics FEM analysis.


12/65



Global minimum of a scalar function

Global minimum : f(x*)


13/65



Optimality conditions Necessary condition: f(x) = 0 Sufficient condition: f(x) > 0 This will identify only local optima.

For general functions,there are no conditionsthat will ensure a global

optimum.

If f(x) is continuous in a closed and limited set, S, then f has aglobal minimum in S ( Weierstrass theorem)


14/65



Positive definite matrices

Let d be an arbitrary direction vector at point x . Thequadratic form d T H(x)d is said to be positive definite,positive semidefinite, negative semidefinite, negativedefinite if d TH(x)d > 0, 0, 0, < 0, respectively, for all d 0 at x.

If d T H(x)d can assume positive as well as negativevalues, it is said to be indefinite.

If d T H(x)d is positive definite, positive semidefinite, etc.,then matrix H(x) is said to be positive definite, positivesemidefinite, etc .


15/65



Optimality conditions for multi-dimensional problems


16/65



Stationary points

When the stationary point is a localmaximum? When the stationary

point is a saddle point? When the stationary

point is a local minimum?


17/65



When can we be sure to find an optimum A convex function is one that has positive definite

Hessian everywhere. A convex function has only one minimum

the global one . In 1D a convex function is one that

everywhere has positive second derivative .


18/65



Convex Sets

S is a convex set when for any pair of points,(x 1,x 2 ) belonging to S, a straight line connectingx 1 and x 2 will be completely contained in S.

This applies in any number of dimensions.


19/65



Convex Optimization A convex optimization problem: If the objective function

is convex and the feasible domain is a convex set then theoptimization problem is convex.

If all the constraint functions are convex, then the feasible

domain is convex . Convex optimization problems have onlyone optimum - the global one.

For a convex optimization problem, if we find a stationary

point, then that will also be the global optimum. Thenecessary conditions are also sufficient. Most optimization algorithms are for convex optimization.

For functions defined implicitly (as in FEM calculations),then we cannot easily check if the function is convex.

Linear problems are always convex.


20/65



Optimization as an analysis tool Many physical phenomena are based on optimality.

For example, we already shown that equilibrium states instructures and solids can be computed by minimizing thetotal potential energy .


21/65



Minimization of potential energy with contact But what if you had some constraints on the

displacements X 1 and X 2 ? (e.g. contact constraints)

You will now need to solve this problem usingoptimization techniques.


22/65



1D Minimization Problem Golden Search Method

We assume that a function f(x) is given, and we want tofind its minimum in [A,B].

We also assume that is expensive to compute f(x). We must find the minimum with the least possible number

of function evaluations. The function is implicit we dont

know what the graph looks like. The golden section search is atechnique for finding the minimum by

successively narrowing the range of values inside which the extremum is known to exist.

The algorithm maintains the function values for triples of

points whose distances form a golden ratio.
http://en.wikipedia.org/wiki/Golden_section_searchhttp://en.wikipedia.org/wiki/Golden_section_search


23/65



Golden Search Method The diagram illustrates a single step in the

technique. The value of f (x ) has already beenevaluated at x 1, x 2, & x 3. Since f 2 is smaller thaneither f 1 or f 3, a minimum lies inside the interval

[x 1,x 3]. In the next step in the minimization process, we

probe the function by evaluating it lets say at x 4.

We choose x 4 somewhere inside the largestinterval, i.e. between x 2 and x 3. From the diagram, it is clear that if the function yields f 4a then a minimum

lies between x 1 and x 4 and the new triplet of points will be x 1, x 2, and x 4. If the function yields the value f 4b then a minimum lies between x 2 and x 3,

and the new triplet of points will be x 2, x 4, and x 3. In either case, we construct a new narrower search interval that is

guaranteed to contain the function's minimum .
http://upload.wikimedia.org/wikipedia/commons/5/52/GoldenSectionSearch.png


24/65



Golden Search Method The new search interval will be either

between x 1 and x 4 with length a +c , or between x 2 and x 3 with a length of b . Thegolden section search requires that theseintervals be equal since otherwise the methodcould lead to the wider interval being usedmany times, thus slowing the rate of convergence. To ensure that b = a +c , thealgorithm should choose x 4 = x 1 x 2 + x 3.

The golden section search chooses the spacing between these points insuch a way that these points have the same proportion of spacing as thesubsequent triple x 1,x 2,x 4 or x 2,x 4,x 3. By maintaining the same proportion

of spacing throughout the algorithm, we avoid a situation in which x 2 isvery close to x 1 or x 3, and guarantee that the interval width shrinks by thesame constant proportion in each step.
http://upload.wikimedia.org/wikipedia/commons/5/52/GoldenSectionSearch.png


25/65



Golden Search Method Mathematically, to ensure that the spacing

after evaluating f (x 4) is proportional to thespacing prior to that evaluation, if f (x 4) is f 4a and our new triplet of points is x 1, x 2, and x 4then we want:

c/a=a/ b

However, if f (x 4) is f 4b and our new triplet of

points is x 2, x 4, and x 3 then we want:

c/(b-c)=a/b

Eliminating c from these two simultaneousequations yields: b/a= = =1.618where is the golden ratio .

( )1 5 / 2+
http://en.wikipedia.org/wiki/Golden_ratiohttp://upload.wikimedia.org/wikipedia/commons/5/52/GoldenSectionSearch.pnghttp://en.wikipedia.org/wiki/Golden_ratio


26/65



Golden Search Method

Each iteration removes 1 - 0.618 = 38% of theinterval.

After n iterations, the interval is reduced to 0.618 n

times its original size. If n is 10, less than 1% of the original interval

remains. If n=15, less than 1 remains. The algorithm is stable but requires that thefunction is unimodal (has one minimum) in the

interval.


27/65



The bisection method

If we can compute the gradient of thefunction , then we know to which side of acomputed value, the function decreases.

Then, we can cut the interval in half andobtain faster convergence.


28/65



The bisection method

After 10 iterations, about 1 of the interval is left. We need to have gradient information which

assumes that the function is differentiable which

was not the case with the Golden section method. This method is less robust than

golden section.


29/65



Polynomial interpolation

We compute the function values in the end pointsand a point in the middle. We fit a parabola throughthe three points.

We then analytically determine the minimum of theparabola. We let the new point replace

the worst of the previousones and repeat until

convergence.


30/65



Polynomial interpolation

Convergence is very fast for smooth convexfunctions (2 nd order differentiability is required)

No gradients required.

Only one function evaluation is needed for eachnew iteration

The algorithm may divergecompletely.


31/65



Unconstrained minimization in multiple dimensions

Choose a search direction, d(k)

Minimize along the search

direction (e.g. by goldensection, line search, etc). Step in the search

direction by a (k) d (k).

Update and repeat untilconvergence


32/65



Steepest Descent

As a search direction, choosethe path that goes as muchdownhill as possible.

This algorithm is known assteepest descent .


33/65



Steepest Descent

Two consecutive steepest descent directions areperpendicular to each other.

The algorithm approaches the optimum using only

very few directions. The steps in each direction get

smaller for each iteration. However, convergence in general

is slow.

S D


34/65



Steepest Descent

On problems with similar scales in the differentvariable directions, steepest descent often workswell.

If the level curves are circular, then the optimum isfound in the first chosen direction. Otherwise, the algorithm typically requires many

iterations.

Th j di h d


35/65



The conjugate gradient method The search direction is here computed by the formula:

The gradient vanishes at the optimum.If the moves towards the minimum are

going well, then the gradient gets smaller for each iteration. In this case, b (k) issmall and the conjugate gradient method does not provide

much correction from the steepest descent method. However, if the gradient does not get smaller, we obtain

significant correction from what the steepest descent

method does.

P l M h d


36/65



Penalty Methods You already have used these methods in your

homework! Penalty methods replace the original constrained

problem with an equivalent one without constraints. The transformed problem is solved as an unconstrained

optimization problem.

E i P l M h d


37/65



Exterior Penalty Methods This penalty does not come into play until a constraint

has been violated. The severeness of the penalty depends on the penalty

factor, r. Small values of r will cause constraint violations . Large

values will make the problem difficult to solvenumerically.

The range of r values is problem dependent.

E t i P lt M th d


38/65



Exterior Penalty Methods Consider the following problem: Linear

objective function and two constraints intwo dimensions.

Consider now the penalized problem withr = 0.05. The optimum

falls far from thesolution to the

original problem.

Optimum

E t i P lt M th d


39/65



Exterior Penalty Methods Penalized problems, r =0.1 and r = 1.0. For r=1 , the

optimum approachesthe solution to the

original problem butnever reaches itcompletely.

For r=1, the levelcurves getsharper edges andthe problem becomesmore difficult to solve

numerically

E t i P lt M th d


40/65



Exterior Penalty Methods The penalty term becomes active only after constraint

violation The objective function inside the feasible domain is not

affected

The penalty objective function is defined everywhere andwe dont need a feasible point to start the process.

The solution always falls slightly outside the feasible

domain of the original problem (where the original problemmay be undefined).

Increasing the penalty makes the problem more difficult to

solve numerically. The method handles equality and/or inequality

constraints .

I t i P lt M th d


41/65



Interior Penalty Methods This penalty is always present but it

becomes more important when aconstraint is approached.

The penalty goes to infinity at the constraint. The penalty isminus infinity right outside the constraint.

The penalty depends on the parameter r. For small r, the constraints kick late

but suddenly as we approach them. The penalty is always present. If a constraint is violated, we may

never return to the feasible domain. Weneed to start from a feasible point.

The solution falls inside the feasible domain of the originalproblem. Thus all solutions are usable.

For higher r, the problem more difficult to solve numerically.

The algorithm handles only inequality constraints.

1

1( ) ( )

( )

m

i i f r g =

= + x x

x

Linear programming


42/65



Linear programming

If the functions g i , i=0..n are linear, we callthe solution of such problems linear programming

Linear Programming


43/65



Linear programs are convex and usually have one optimum.

The feasible domain is apolyhedron in n dimensions.The optimum is always in a corner

of the polyhedron. If n = m, then our problem

degenerates to a system of equations with usuallyone solution.

If n < m, then it usually has no solutions. If n > m, the feasible domain has infinitely many points, and we

have an optimization problem. There is no limit on the number of inequality constraints.

Linear Programming

Linear Programming


44/65



Linear Programming

Slack variables convert inequalities to equalities

By increasing the number of variables, we get ridof all inequality constraints except the ones thatrequire the slack variables to be positive .

Linear Programming Standard Formulation


45/65



Linear Programming- Standard Formulation

The standard linear programming problem:

In matrix form:

The Simplex Algorithm


46/65



The Simplex Algorithm The solution is always in a corner of the polyhedron.

Consider that we have 2 variables and 5 inequality constraints (as inthis figure).

We add slack variables to obtain 7 variables and 5 equality constraints.

Each corner is now the solution to a 5-subset (the basis set) of the 7 variables.

The simplex method steps from onecorner to another by replacing thevariables in the basic set one by onewith variables from the outside .

Investigate the coefficients of the variables inside and outside the basic

set to see if any neighbouring corner is better than the one we are in. If there is a better neighbour, introduce the corresponding variable in

the active set and step to this neghbouring corner. For very large problems, interior point methods can be better.

Search methods for constraint optimization


47/65



Search methods for constraint optimization Find a feasible search direction (not violating the constraints)

Minimize along the search direction Repeat until convergence.

Whenperforming1-D linesearch,we stopwhen we hita constraint.

Choice of feasible directionis a compromise between

- quick reduction- avoiding constraintviolation

Feasible direction


48/65



Feasible direction

This can be posed as follows:

This linear problem can be solved by the simplex method.

It depends on derivatives (sensitivities) only in the search direction. Theline search can be done with a golden section search. If we start from a feasible point, we stay in the feasible domain. Thus

every iteration is feasible and better than the previous iteration.

Sequential Programming


49/65



Sequential Programming In sequential programming we approximate the functions g i ( x ).

The optimization procedure is applied iteratively as a sequence of subproblems. Hence the term sequential programming.

We make a Taylor expansion of the functions from the current point,x (k). If we only include up to linear terms, then the subproblem is linear and can be solved by the Simplex method (sequentiallinear programming)

The approximation is only valid in a certain region ( trust region )

around x(k).

We thus need to constrain the solution (inside what wecall move limits ) of the subproblem to the trust region.

Sequential Linear Programming


50/65






51/65



Sequential Linear Programming Move limits need to be adjusted during the

optimization process.

When a design variable approaches the optimum

from one side, we relax the move limit on x j a little.

If not, we tighten.

This way, the move limits on each variable adjustgradually to the nature of the problem.

Convex Programming


52/65



Convex Programming SLP is an attractive optimization method

because it only require gradients. However, SLP requires move limits and

thus is prone to oscillation.

In addition, in SLP, the approximationto the feasible domain is not conservative(it does not under-estimate the size of

the feasible domain). This is importantas we approach the optimum through asequence of feasible designs.

Convex programming addresses many of these problems.

Sequential Quadratic Programming


53/65



Sequential Quadratic Programming If we use quadratic approximations of the functions, then we get a

much more accurate subproblem.

H is the Hessian matrix containing second derivatives:

Sequential Quadratic Programming


54/65



Sequential Quadratic Programming If H is positive definite, then the approximation curves

upwards and thus increases at some distance from the currentpoint. Thus the problem is automatically bounded, and we donot need move limits.

It is possible to derive linear optimality conditions for aquadratic problem. This means that it can be solved by analgorithm using Simplex as a subroutine.

H contains second order derivatives, and many of them, order n, although it is symmetrical. Real second order sensitivityanalysis is very time consuming.

Second order algorithms are more sensitive to non-smoothness of the functions than lower order algorithms. Thesolution of a QP problem often requires inversion or factorization of H. This is time consuming, if n is large.

Quasi-Newton Methods


55/65



Quasi Newton Methods We start out with a linear approximation and a Hessian H = I.

For each step in the process, we save the computed gradients of allfunctions.

The gradients of multiple design points are used to create an overall

approximation of H. This approximation improves as more iterations are performed.

Quadratic functionapproximated from

gradients.

Quasi-Newton Methods


56/65



Quasi Newton Methods Some methods can approximate the inverse Hessian directly.

If the process does not converge in ~10 iterations, then the overallbehavior of the functions is not nearly quadratic, and we need toreinitialize H to I.

Some functions are notapproximated well by

quadratic forms.

Summary of constrained optimization methods


57/65



Summary of constrained optimization methods

Sensitivity Analysis


58/65



Sensitivity Analysis Optimization methods rely on gradients

If you know the analytical expressions for g i, then you candifferentiate analytically ( analytical method )

If gi is numerically (e.g. FEM) defined, then you can compute the

gradient numerically ( use finite differences ). If you know something about g i but not the full expression, then it

may be possible to work in part analytically ( semi-analyticalmethod )

Finite Difference Approximations


59/65



Finite Difference Approximations

Choosing the right value of

x is acompromise between round-off errors and truncation errors.

Forward finite difference

Backward finite difference

Central finite difference

Finite Element Analysis


60/65



Finite Element Analysis In finite element analysis, we roughly go through the following steps:

If we denote the size of the problem in terms of nodes, elements or degrees of freedom by n, then step 1 is proportional to n, step 3 is propotional to n 2 , and step 2 is proportional to n 3 .

In FE analysis, we solve the problem: Ku=F We imagine that the problem depends on design variables x = {x1,

x2, .. , x n }, and we want to find the sensitivity w.r.t. x j.



61/65




We are solving a system with the same [K] as before, sowe do not have to perform the time-consuming operation[K] =[L][D][L]T again!

The finite differencescomputation of these terms is notvery time consuming. It is only

proportional to the problem size.

:[ ]{ } { } Direct FEM Analysis K u F =

N

{ } { } [ ]:[ ] { }

j j j

Unknown Pseudo load

sensitivity vector vector for sensitivityanalysis

u F K Sensitivity FEM Analysis K u

x x x

=

{ } { }

j j

F F

x x

[ ] [ ]

j j

K K x x



62/65



Sensitivity Analysis Let us consider the following simple problem: study the

sensitivity of the end deflection w.r.t. the beam length. This problem is solved within the Bernoulli-Euler theory

using the FEM analysis considered earlier. It is also very simple to compute the analytical solution to

this problem.



63/65



Sensitivity Analysis It turns out that the error in the computed sensitivity grows with the

square of the number of elements! Recall that in the FEM formulation we have both displacements and

rotations in the u vector and both forces and moments in the f vector. The resulting tip deflection is the difference

between the contributions from all the forcesin different directions. The more elements,the more forces.

If all the forces are a little bit wrong, then theresulting deflection is a small difference between two large erroneousnumbers

Various ways to eliminate this problem: (a) Design-differentiate theoriginal governing equations and then use FEM discretization; (b) Donot introduce FD errors until after the equation solution has been done,etc.

{ } { } [ ][ ] { }

j j j

u F K K u

x x x

=

Response Surface Methods


64/65



p Sensitivity analysis can be a difficult task

Shape sensitivities are often FEM mesh-dependent. Sensitivities may not be defined if you have non-smooth conditions (e.g.

contact, friction, etc)

You may not have access to the direct FEM analysis. Not easy to include experimental data or prior information into thesensitivity formulation.

In these cases we can use a response surface method it only requires

access to direct analysis (as a black box) In response surface methods you run the direct analysis for various

values of the design variables and then use some fitting software for theresponse of the system ( build a hypersurface in the design space ). Thishypersurface can then be explored with any optimization techniques.

Curse of dimensionality : Unfortunately, the number of direct simulations(which are usually very expensive in the context of FEM) needed growsvery fast with the number of design variables.

Topology Optimization


65/65

p gy p

Compute the density of thematerial.

http://www.topopt.dtu.dk

Documents

FEM Zabaras ComputationalDesign