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Developments on Shape Optimization at CIMNE
October 2006
www.cimne.comAdvanced modelling techniques for aerospace SMEs
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Geometry:B-spline. Definition points r(i)
Geometry:B-spline. Definition points r(i)
Shape parametrization
Design variables:Coordinates of some definition points Design variables:Coordinates of some definition points
B-spline expression:in terms of the coordinates of “polygon definition points” ri.
B-spline expression:in terms of the coordinates of “polygon definition points” ri.
Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))
Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))
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Shape parametrizationDesign variables:shape parameters (example of FANTASTIC ship hull)Design variables:shape parameters (example of FANTASTIC ship hull)
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Design variables:
deformation of patches defined with a C1 continuity interpolation function over the bulb of a ship hull
Design variables:
deformation of patches defined with a C1 continuity interpolation function over the bulb of a ship hull
Shape parametrization
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Mesh generation and quality aspects
Shape optimization problem:
f objective function
x vector of design variables
g set of restrictions
Deterministic methods
Evolutionary algorithms
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1. Total computational cost of optimizationclosely related to FE analysis cost per design.
2. Bad quality of FE analysis:
Introduce noise in the convergence
Possible bad final solution.
Evolutionary methods involves the analysis (FEM) of many different designs.
Influence of mesh generation:
Mesh GenerationMesh Generation
Mesh generation and quality aspects
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Classical strategies for meshing each individual:
1. Adapt a single existing mesh to all the different geometries.
Existing strategies allow adapting an existing mesh for very big geometry modifications preventing too much distortion.
Cheapest strategy
No control of the discretization error.
2. Classical adaptive remeshing for the analysis of each design.
Good quality of results of each design
High computational cost (each design is computed more than once)
Mesh generation and quality aspects
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Adaption of a mesh to the boundary shape modifications
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Representativeof population.Representativeof population.
Generation of an adapted mesh for each design in one step using error sensitivity analysys
Mesh adaptivity based on Shape sensitivity analysis
Mesh adaptivity based on Shape sensitivity analysis
Projection parameters (sensitivity of nodal coordinates
and error indicator)
Projection parameters (sensitivity of nodal coordinates
and error indicator)
Final h-adapted mesh of representative
Final h-adapted mesh of representative
h-adaptive analysis of
representative
Classical sensitivity
analysis
Projection to individuals
h-adapted mesh for 1st individual
h-adapted mesh for 1st individual
h-adapted mesh for 2nd individual
h-adapted mesh for 2nd individual
h-adapted mesh for 3rd individual
h-adapted mesh for 3rd individual
h-adapted mesh for Pth individual
h-adapted mesh for Pth individual
in “one-step” !!
Low cost control of discretization errorLow cost control of discretization error
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Geometry:B-spline. Definition points r(i)
Geometry:B-spline. Definition points r(i)
Parameterization of the problem
Sensitivity analysis of the system of equations:
Sensitivity analysis of the B-spline expression:
Design variables:Coordinates of some definition points Design variables:Coordinates of some definition points
B-spline expression:in terms of the coordinates of “polygon definition points” ri.
B-spline expression:in terms of the coordinates of “polygon definition points” ri.
Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))
Polygon definition points vector, R:Obtained solving V=NR(V imposed conditions at r(i))
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Mesh generation and mesh sensitivity
Mesh Generator
Advancing front method
Background mesh defining the size δ at each point.
Mesh sensitivitySmoothing of nodal coordinates
Mesh Sensitivity
Boundary nodal points: obtained by the B-spline sensitivity analysis.
Internal nodal points: spring analogy (fixed number of smoothing cycles)
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Finite element analysisSolution of standard elliptic equations
Discretization:
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Error estimationEstimation in energy norm of the error: ZZ-estimator
Stress recovery: Global least squares smoothing
Approximation of total energy norm:
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Sensitivity analysis of the error estimator
Discrete-Analytical method:
Discretized model (element integral expressions) are analytically differentiated
with
Sensitivities of - displacements- strains - stresses
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Sensitivities of smoothed stresses:
Sensitivities of error estimator:
Sensitivities of the strain energy:
Sensitivity analysis of the error estimator
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The used evolutionary algorithm
Parameter vector of i-th individual
of generation t
For each individual, a new trial vector is created by setting some of the parameters up
j(t) to:
Parameters to be modified and individuals q, r, s are randomly selected
The new vector up(t) replaces xp(t) if it yields a higher fitness.
Non accomplished restrictions integrated in objective function using a penalty approach.
Evolutionary algorithm: classical Differential Evolution (Storn & Price).Evolutionary algorithm: classical Differential Evolution (Storn & Price).
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Projection to each design and definition of the adapted mesh
Representative of populationRepresentative of population pth individual of populationpth individual of population
Projection using shape sensitivity
analysis
Projection using shape sensitivity
analysis
Mesh coordinates
Error estimation
Strain energy
Generation of h-adapted mesh.
Admissible global error percentage
Mesh optimality criterion: equidistribution of error density
Target error for each element
New element size
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Pipe under internal pressure1
4
3
2
x
y P
4 design variables
Circular internal shape
P=0.9 MPa
vm 2 MPa
||ees|| < 1.0%
30 individuals/generation
Design variable
Initial Value
Data Range
Constraints
V1 20 [ 5.2 − 50.0 ]
V2 19 [ 4.0 − 50.0 ]
V3 19 [ 4.0 − 50.0 ] V3 < V1 − 0.5
V4 20 [ 5.2 − 50.0 ] V4 < V2 + 0.5
Optimal analytical solution for external surface:
• Circular shape Ropt = 10.66666
• Cross section area Aopt = 69.725903
Optimal analytical solution for external surface:
• Circular shape Ropt = 10.66666
• Cross section area Aopt = 69.725903
Minimize unfeasible designs
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Analytical Optimal shape
A = 69.725903
Optimal shape obtained
(B-spline defined by 3 points)
A = 70.049
0
Pipe under internal pressure
01 -234567891 0111 21 31 41 51 6 -1 71 81 92 0 -2 12 2 -2 32 4 -2 52 6 -2 72 8 -3 03 1 -3 53 6 -4 24 3 -5 55 65 7 -6 56 66 7 -8 18 28 3 -9 19 2 -9 59 6 -9 89 9 -1 0 31 0 4 -1 2 41 2 5 -1 2 71 2 8 -1 8 5185 generations
30 individuals/generation
only 3% individuals required additional remeshing
185 generations
30 individuals/generation
only 3% individuals required additional remeshing
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Pipe under internal pressure
0
100
200
300
400
0 50 100 150Generation
Are
a
Minimun = 69.725903
0.1
1
10
100
1000
0 50 100 150Generation
Err
or %
0.46%
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Gravity Dam
1
4
3
21 0
7
6
8
9
5
3
P la n e s tra in
= 2 3 0 0 k g /mE = 1 3 .1 1 0 N /m
= 0 .2 5
x 1 0 2
Optimization of internal boundary
10 desing variables
vm 2.75 MPa
||ees|| < 3.0%
30 individuals/generation
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Gravity Dam
001234 -56 -7891 0111 2 -1 31 2 -1 31 41 51 61 71 81 92 0 -2 12 2 -2 42 52 62 7 -2 82 9 -3 03 1 -3 53 6 -3 94 04 14 2 -4 34 4 -4 64 7 -4 95 05 15 2 -5 65 7 -6 56 6 -6 76 8 -7 87 9 -9 39 4 -1 0 911 0 -1 2 1
Original Individual
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Optimizedshape
Originalshape
Gravity Dam
5800000
5900000
6000000
6100000
6200000
6300000
6400000
6500000
6600000
6700000
6800000
6900000
0 20 40 60 80 100 120Generation
Are
a
Original Individual
120 generations
30 individuals/generation
only 5% individuals required additional remeshing
120 generations
30 individuals/generation
only 5% individuals required additional remeshing
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Gravity Dam
Average Individual in Generation 28
Reference mesh
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Fly-wheel
FE model of Initial design space Optimum topologyInitial design space
Initial model for further optimization (60 design variables)
8 independent design variables
60 design variables
8 independent design variables
vm 100 MPa
||ees|| < 5.0%
15 individuals/generation
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Fly-wheel
O rig . In d iv. 1-34 -67 -2 42 5-4 64 7-8 88 9 -9 29 39 4-12 71 28 -16 71 68 -17 71 78 -1 8 81 89 -20 12 02 -2 0 62 07 -2 3 42 35 -2 4 52 46 -25 92 60 -29 72 98 -3 0 0 Original
Design
OptimumDesign
1.441.451.461.471.481.491.501.511.521.53
0 50 100 150 200 250 300Generation
Wei
ght i
n kg
300 generations
15 individuals/generation
Weight reduction 1.53 1.445 kg
(0.25 0.17 in the design area)
(Deterministic: 1.53 1.45 kg)
300 generations
15 individuals/generation
Weight reduction 1.53 1.445 kg
(0.25 0.17 in the design area)
(Deterministic: 1.53 1.45 kg)
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Conclusions
A strategy for integrating h-adaptive remeshing into evolutionary optimization processes has been developed and tested
Adapted meshes for each design are obtained by projection from a reference individual using shape sensitivity analysis
Quality control of the analysis of each design is ensured
Full adaptive remeshing over each design is avoided
Low computational cost (only one analysis per design)
Numerical tests show
• The strategy does not affect the convergence of the optimization process
• Good evaluation of the objective function and the constraints for each different design is ensured
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Thank you very much
