Derivatives of static response from linear finite element analysis

1. Derivatives of static response from linear finite element analysis • Local search algorithms benefit from derivatives even when they are calculated by finite differences • Often derivatives can be calculated at fraction of cost of finite-difference derivatives • Goal of today’s lecture is to show why this is usually true for static response • Key concepts will be the direct and adjoint variants of derivative calculation • Also there are some numerical pitfalls

2. Properties of governing equations • Equations of equilibrium in finite element system: Ku=f • K is normally large and sparse and ill-conditioned, but positive definite • A variant of Gaussian elimination is used for solution, without pivoting(averts matrix fill-up), that allows most of the computation to be done without involving the load f. • In aerospace applications system needs to be solved repeatedly for thousands of different right hand sides. Why?

3. Cholesky decomposition • A positive-definite stiffness matrix K can be decomposed as: K=LTDL, where L is a lower triangular matrix with 1s on diagonal, and D is a diagonal matrix • Solution of Ku=f is replaced by • Why is that good for multiple RHS? • When K has a banded or skyline sparseness structure, it is preserves by L.

4. Direct method • Equations for displacement vector Ku=f • Response quantity of interest g(u,x) • Differentiate • RHS of derivative equation called pseudo load • Can be calculated outside of finite element program • Requires only forward and backward substition

5. Adjoint method • Rewrite derivative equation • For both adjoint and direct method need pseudo load vector and z vector.

6. Direct vs. Adjoint method • Direct method • requires psuedo load calculation and solution for each design variable • minimal effort with new constraints • Adjoint method • requires solution for each new g, and pseudo load calculation for each variable • How do we choose? • See beam example in Section 7.2

7. Problems direct & adjoint • For a dense matrix, the Cholesky decomposition requires about n3 operations, and forward or backward substitution about n2 operations. Estimate the operation count for calculating the derivatives of 3 displacement components with respect to 10 design variables for n=1000, using the direct and adjoint methods. Assume that the pseudo load calculation is negligible. • How do the two methods compare for calculating the derivative of the compliance, uTf?

8. The semi-analytical method • Analytical derivatives of stiffness matrices are burdensome especially in commercial software. Why? • Most resort to finite-difference calculation of derivatives of stiffness matrix and force vector

9. Problems with shape derivatives for a car model, circa 1985 . Derivative of compliance with respect to a dimension of car

10. Cantilever beam example

11. Difficulty and partial solution • We live dangerously when we solve ill-conditioned equations like FE equations • Loading that excites huge errors are not met in real life, they look like high vibration modes • The pseudo load can come to resemble such loads because it is not derived from physical loading • Haftka, R.T., “Stiffness-Matrix Condition Number and Shape Sensitivity Errors,” AIAA Journal, Vol. 28, No. 7, pp. 1322-1324, 1990. • Van Keulen refined semi-analytical method purges K of rigid body component as a way around most common manifestation of problem

12. Problem semi-analytical • Describe in 45-55 words the pros and cons of the semi-analytical method • Use the semi-analytical method to reproduce the results of the figure on slide 10. You will need to generate the stiffness matrix of a cantilever element with several elements. Check that you assembled the right matrix by noting that the tip displacement of a beam under an end load P is • If you find it difficult to do the problem in non-dimensional form, you may use the following data: P=1,000 lb, L=120”, E=30 msi, I=3 in4.