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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
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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
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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