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1 © Dassault Systèmes Ι Confidential Information Abaqus Topology and Shape Optimization Module

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Abaqus Topology and Shape Optimization Module

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Contents

IntroductionOptimization workflowSupported functionalityExamples

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Topology and Shape Optimization

Topology optimization: Modify stiffness (by modifying material) Good for evolving optimum shape

Shape optimization: Moves nodes Good for fine tweaking of shape

Both support: Contact Geometric non-linearity Nonlinear materials:

o Within the design area: *plastic, *hypoelastic, most *hyperelastico Outside design area: all

Manufacturing restrictions

Export smoothed shape to STL or INP

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Optimization Workflow

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Topology optimization Modify stiffness

Shape optimization Move nodes

5-50 solver iterations is typical

Afterwards, run “Smooth”, then export INP or STL to CAD

Currently Abaqus/Standard only

Specify problem

Modify .inp file

Standard

Postprocess

Final Solution ?

No

Visualize Smooth output

Export to CAD

Write .inp file

Shape or Topology Optimization components

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ATOM Workflow: Setup

The flow chart on the left shows the user actions required to setup the optimization

Each user action is associated with a manager in the Optimization module accessible from the Optimization Module Toolbox or the Model Tree

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ATOM Workflow: Execution and Monitoring

Once an Optimization Task is setup, an Optimization Process needs to be defined to execute the optimization

Users may have multiple Abaqus models and optimization tasks defined. An optimization process refers to a unique Model and Task combination.

Right-click on the optimization process to access: Validate, Submit, Restart, Monitor, Extract and Results postprocessing

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ATOM Workflow: Results

The Abaqus Visualization module allows for convenient visualization of optimization results Postprocessing will be discussed in detail in subsequent lectures

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ATOM Workflow: Extraction

Optimization results can be smoothened and extracted as Abaqus input files or STL files

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Supported functionality overview

Design Responses provide variables for the optimization solverObjective Functions define how thoseDesign Responses should be used (sum/min/max/formula/etc)Constraints determine bounds for the optimization solverGeometric Restrictions provide for manufacturingStop conditions

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Supported Functionality

Single or multiple termsRegion basedOperators: Sum Minimum Maximum Deviation from Max Number of values

Select the step to extractresults from or load cases

e.g: “sum the elementstrain energy”

Design Responses

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Supported Functionality

Uses already defined Design ResponsesAllow combiningmultiple DesignResponsesTargets: Minimize Maximize Minimize the maximum

weighted difference fromthe maximum

E.g: “minimize Design Response 1”

Objective Functions

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Supported Functionality

Uses already defined DesignResponsesAllows constraining the DesignResponse to: Greater than Greater than a fraction of the initial

value Less than Less than a fraction of the initial value

E.g: “Constraint the volume to be less than 35% of the original volume”

Constraints

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Supported Functionality

The following geometric restrictions are available: Frozen areas Member Size

o Min, max, envelope Cyclic symmetry Planar, Point and Rotational Symmetry Demolding

Geometric Restrictions

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Supported Functionality

New job super-type (similar toAdaptivity or Co-execution)Once a job is submitted, CAEcan be closed.Restarting a failed analysis runis supportedAllows control onmaximum number ofjobs, results ODBmerge, etcQueues from CAEare supported

Job control

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Supported Functionality

Standard post-processing of resultsAdditional support for viewing the generated iso-surfaceExtraction of the results surface to STL or Abaqus input file Allows export to CAD or back to

Abaqus/CAE for rerunning withATOM’s Shape Optimization

Post processing

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ATOM Example : Pull Lever on a Press

Lever is redesigned to reduce stresses, with minimum weight

Initial volume

FEA model

Proposal

Design

Validate

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ATOM Example: Bracket with Loads

Bracket with loads, some torsion and press fitsFinal result is sensitive to applied loads/etc

Original Geometry Demold Demold with thickness control

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SIMULIA’s Design Exploration and Optimization Tools

ATOM Isight

Tuned for topology and shape optimization

A general purpose design exploration and optimization package

Non parametric Parametric

Can handle a very large number of design variables. (~100K-1000K)

Meant for small number of design variables(~10-100)

Multiple objectives are summed up to single objective

Multi-objective, multi-discipline optimizations possible

ATOM Isight

DOE

Monte Carlo

Exploration

Taguchi RD

Six Sigma

OptimizationTopology optimization

Shape optimization

Test DataMatch

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Demo- Topology Optimization of a Gear

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More on Topology Optimization

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Topology Optimization

Consider a beam structure clamped at one endTypically the goal of a topology optimization is to produce a landscape that minimizes an objective given a certain amount of material removal In this case we want to minimize the tip displacement of this beam

Therefore, we define our Objective and Constraint Objective: Minimize displacement uout

Constraint: Reduce mass by a given amount

Design domain: All elements in the beam structureDesign variables: Relative density of each element in the design domain

Load

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Problem Statement

The problem statement is the following:

where:ve is the element volume

re is the element relative density

n is the number of elements in the optimization domain

Load

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Topology Optimization as an Integer Value Problem

The Topology optimization problem can be posed as an integer value problem The design variables can only take the values 1

or 0 This results in voids and fully dense elements

only, which is favorable.

It is indeed possible to solve the integer value problem

However, such approaches are quite expensive computationally and therefore impractical

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Continuous Design Variables

In order to apply gradient-based optimization techniques (which can be more efficient), the integer value problem is relaxedThe design variables (relative densities) are assumed to be continuous

How do we interpret the intermediate density elements? We don’t! We use an approach that penalizes

intermediate density elements so that they are not favorable in the final solution.

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SIMP

How do we avoid intermediate density elements in the final solution? By using the Solid Isotropic Material with Penalization method (SIMP) for

material interpolation This popular method represents the stiffness matrix as:

where: Ke is the element stiffness matrix at the global level

rmin is a small relative density

re is the element relative density

p is a penalty term N is the number of elements in the domain

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SIMP

SIMP serves several functions:1. Material interpolation

o Assigns stiffness to intermediate density elements

2. Penalization of intermediate elementso Makes the intermediate density elements unfavorable in the optimization by making the

stiffness of the intermediate density elements small compared to the volume

3. Avoids singular stiffness matriceso Doesn’t allow the stiffness matrix to become singular by introducing a very small

amount of minimum relative density

o Note: rmin need not be introduced in SIMP (next slide)

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SIMP for Nonlinear Analysis

The basic statement for equilibrium in a nonlinear analysis is that the external forces P and the internal forces I must balance each other (within a tolerance) at every node; i.e., P - I = 0 That is, equilibrium is established when the force residual R is “zero”

We can write the topology optimization problem as:

• The material tensor is scaled by SIMP• For simple material models, such as linear

elasticity, the scaling would be the following:

• This scaling becomes more complicated for advanced material models; please see the ATOM documentation for a list of supported materials.

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RAMP

RAMP (Rational Approximation of Material Properties) is an alternative to SIMP RAMP also penalizes the intermediate density elements

where:Ke is the element stiffness matrix at the global level

rmin is a small relative density

re is the element relative density

p is a penalty term N is the number of elements in the domain

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The Sensitivity Based Optimization Algorithm

Gradient-based optimization algorithms are preferable for solving this class of problem. They are efficient But they can get trapped in local optima

In particular, the Method of Moving Asymptotes (MMA) is preferred This method, like SLP and SQP, works with a sequence of approximate sub-

problems to arrive at the optimum solution It is appropriate for problems with very large numbers of variables

Gradient-based methods require sensitivity information Calculated by adjoint methods Optimization performs a linear perturbation to calculate sensitivity

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Stiffness and Controller Based Topology Optimization

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Stiffness

Often the goal of Topology optimization is to find the stiffest possible structure given a certain amount of material removal. For example, assume you are trying to reduce the weight of the

component shown on the right. o You know your structure is going to be more compliant as a result of the

material removal. o You want to find out what landscape will give you the most stiffness (or

least compliance) given the material removal, and evaluate if this new landscape is stiff enough to do the job your component was designed to do.

o You can use controller-based topology optimization to produce this new landscape (bottom right)

• Note: You may also use sensitivity-based optimization to minimize compliance (max stiffness) but the controller-based approach has certain advantages (discussed later)

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Stiffness

We have a scalar measure of the compliance of a structure Compliance is given by UTKU

o Here K is the global stiffness matrix and U is the global displacement vector

Thus, we may write the stiffness optimization problem as:

Note: In ATOM, compliance is selected as the sum of the strain energies of the elements in the design area

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Controller Based Optimization for Stiffness Optimization

The controller in ATOM is an extension to the optimality criteria method for topology optimization

Optimality criteria methods have been shown to efficiently produce good results for stiffness optimization (with volume constraints)

TIP: when solving a complex topology optimization problem, try the controller based optimization (of the reduced problem) first.

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Controller Based Optimization vs Sensitivity Based Optimization

Controller Based General Sensitivity Based

Objective: Strain energy Objective: Strain energy, Volume, Weight, Displacement, Rotation, Eigenfrequency, Reaction

force, Reaction Moment, Internal force, Internal moment, Center of gravity, Moment of Inertia

Constraint: volume Constraint: Strain energy, Volume, Weight, Displacement, Rotation, Eigenfrequency, Reaction

force, Reaction Moment, Internal force, Internal moment, Center of gravity, Moment of Inertia

Solution is guaranteed for a stiffness problem

Solution is not guaranteed for objectives other than stiffness

Doesn’t need to perform a linear perturbation step for sensitivity

calculation

Needs to perform a linear perturbation step for sensitivity calculation

Needs around 15~20 iterations for most problems

Needs around 50~80 iterations for most problems