Developments of a Discrete Adjoint Structural Solver for Shapeand Composite Material Optimization

Marc Schwalbach1 ,Tom Verstraete2, Nicolas R. Gauger3

AD 2016September 15, 2016

Oxford, UK

1von Karman Institute for Fluid Dynamics2Queen Mary University of London3Technische Universität Kaiserslautern

I C++

I Eigen, PETSc, SLEPc

I AD tool CoDiPack

2 a

I FEM linear stress analysis

I FEM vibration analysis

I extension to composite (shell) elements

3 a

linear stress analysis

1. input x ∈ Rn

2. linear system solve A(x)u = b(x)

3. output σmax(u(x)) ∈ R

x(1) =∂σmax∂x

?

4 a

5 a

6 a

b(1) = A−Tu(1)

A(1),i,j = −ujb(1),i

7 a

Ab(1) = u(1)

A(1),i,j = −ujb(1),i

8 a

9 a

rotating beam. ‖ ∂σmax∂x

‖ ∈ R90

10 a

11 a

DOFs ≈ 30k ≈ 200k 200k, RealReverseIndexrelative run time 2.5× T 2.1× T 2.8× Trelative peak memory 7.7×M 6.4×M 5.8×M

gradient evaluation (forward + reverse run). M : peak memory of primal. T : run time of primal

12 a

run time breakdown. 200k DOF axial fan. primal reverse AD

13 a

14 a

15 a

vibration analysis

1. assemble mass matrix B(x)

2. solve generalized eigenvalue system: (A− λiB)ui = 0

x(1) =∂λi∂x

?

16 a

17 a

A(1),ij = λ(1),kuk,iuk,j (1)

B(1),ij = −λ(1),kλkuk,iuk,j (2)

18 a

19 a

rotating beam. ‖ ∂λ0∂x‖ ∈ R90

20 a

21 a

run time breakdown. 200k DOF axial fan. primal reverse AD

22 a

I same linear and eigenvalue solver

I extended design variables x∗ = (x,V )

I lamination params

V = V (t,θ)

24 a

I different stiffness and mass matrices Ac(x∗), Bc(x

∗)

I different failure criterion σTW

∂σTW∂x∗

,∂λk∂x∗

25 a

26 a

27 a

composite flat plate. eigenmode 0 eigenmode 3

28 a

∂λ0∂x

∂λ3∂x

29 a

∂λ0∂V11

30 a

∂λ0∂V11

∂λ3∂V11

31 a

I AD’d linear stress analysis, separate linear solver

I AD’d vibration analysis, separate EV solver

I extension to AD’d composites

32 a

Developments of a Discrete AdjointStructural Solver for Shape andComposite Material Optimization

Marc Schwalbach1

Tom Verstraete2

Nicolas R. Gauger3

financial support

European Commission - IODA4project

1von Karman Institute for Fluid Dynamics2Queen Mary University of London3Technische Universität Kaiserslautern4Industrial Optimal Design using Adjoint CFD 32 a