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Derivatives of static response from linear finite element analysisLocal search algorithms benefit from derivatives even when they are calculated by finite differencesOften derivatives can be calculated at fraction of cost of finite-difference derivativesGoal of todays lecture is to show why this is usually true for static responseKey concepts will be the direct and adjoint variants of derivative calculationAlso there are some numerical pitfalls

Finite difference derivatives cost approximately one additional analysis per design variable. Structural optimization is often done with thousands of design variables, so each time we calculate derivatives by finite differences we will need to perform thousands of analyses.Fortunately, there is a shortcut that allows us to calculate derivatives at a fraction of the cost, which is the topic of todays lecture.

It turns out that there are two ways to reduce the cost of calculating derivatives called the direct and adjoint techniques, and they are the main subject of the lecture. Another important topic are numerical difficulties that may be encountered.1Properties of governing equationsEquations of equilibrium in finite element system: Ku=fK is normally large and sparse and ill-conditioned, but positive definiteA 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?

The shortcut we have for finding derivatives depends on the properties of the stiffness matrix K that we use to solve for displacements from the system Ku=f, with f being the force vector. The matrix is large and sparse, often ill-conditioned, but positive definite.

These properties mean that we usually solve the system of equations by Gaussian elimination, where most of the processing is independent of the force vector. That means that solving for multiple load vectors is efficient, which is important even if we are not calculating derivatives. This is because in aerospace design we often needs to calculate the response to thousands of load vectors coming from different combinations of maneuvers, gust loads, and weight distributions.

The trick to efficient calculation of derivatives will turn out to converting the calculation of a derivative to instead calculate the response to a different load vector.2Cholesky decompositionA 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 matrixSolution 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.

An example of Gaussian elimination is the Cholesky decomposition (Andr-Louis Cholesky (October 15, 1875, Montguyon August 31, 1918, Bagneux) was a French military officer and mathematician). The amount of computation that goes into the decomposition is of order n3, when the matrix is full, and much lower for sparse matrix, but still typically increases with the cube of the dimension. On the other hand the cost of solving for u after the decomposition is complete is of the order of n2. 3Direct methodEquations for displacement vector Ku=fResponse quantity of interest g(u,x)Differentiate

RHS of derivative equation called pseudo loadCan be calculated outside of finite element programRequires only forward and backward substition

Consider the need to calculate the derivative of a constraint function (e.g., the stress at a point) that depends on the displacement vector and a design variable x (such as thickness). When we differentiate the equations Ku=f with respect to the design variable, we will get

Ku+Ku=for Ku=f-Ku=fp

where a prime denotes derivative with respect to x. Note that we found that we can find the derivative u by solving a system of equations with the same stiffness matrix, but with a different load vector f-Ku=fp . This load is called the pseudo load, and its calculation is often inexpensive because the design variable often affects only a small part of the matrix K.

With the direct method of derivative calculation we first solve for the derivative of the load vector and then we substitute it to calculate the derivative of g. The method is most efficient when we have small number of design variables and large number of functions g.4Adjoint methodRewrite derivative equation

For both adjoint and direct method need pseudo load vector and z vector.

The first step in deriving the adjoint method is to substitute the equation for the displacement derivative into the equation for the derivative of g. Then we note that if we apply the inverse of the stiffness matrix to the vector on its left instead of a vector on its right we can get an alternative expression.

The alternative expression involves solving for a load vector equal to z, which is the derivative of g with respect to the displacement components. Since g is usually a stress component, it depends on very few displacement components and the so its computation is usually easy.5Direct vs. Adjoint methodDirect method requires psuedo load calculation and solution for each design variableminimal effort with new constraintsAdjoint method requires solution for each new g, and pseudo load calculation for each variableHow do we choose?See beam example in Section 7.2

Direct method requires solution for one pseudo load for each design variable, but it does not suffer much if there are many functions g. So for example, if we need to calculate the derivatives of many stresses from the derivatives of the displacements, the direct method would be a good choice.

On the other hand, the adjoint method requires a solution for each g. So if we have a single g and a large number of design variables it would be a good method. For example, if we need to calculate the derivatives of one displacement component with respect to many design variables the adjoint method would be ideal.6Problems direct & adjointFor 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?

The semi-analytical methodAnalytical 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

Calculating the derivatives of the stiffness matrix and the load vector with respect to design variables can be burdensome, especially for complex finite element that depend on large number of parameters. These include material properties, dimensions, and the coordinates of the nodes. So it is quite common to calculate the derivatives of K and f by finite differences instead of analytically, and then use them with the direct or adjoint method. The combination is called the semi-analytical method.8Problems with shape derivatives for a car model, circa 1985.

Derivative of compliance with respect to a dimension of car

My PhD student, Bruno Barthelemy, stumbled across a problem with the semi-analytical method when on internship at Ford in 1985. He used the semi-analytical method to calculate the derivatives of the compliance (work of the applied loads) with respect to the dimensions of the car frame. For two of the dimensions he found that it was very difficult to find a step size that provided accurate derivatives.9Cantilever beam example

When Barthelemy came back from his internship, he set to work to find the reason for the problem, and he was finally able to duplicate it for the derivative of the tip displacement of a beam with respect to the length of the beam.

As the number of elements in the beam increased the error of the semi-analytical method increased. This was shown with his own beam finite element code as well as with a commercial finite element code called EAL.

In comparison the overall finite difference approach, where you change the length and recalculate the displacement was very accurate.10Difficulty and partial solutionWe live dangerously when we solve ill-conditioned equations like FE equationsLoading that excites huge errors are not met in real life, they look like high vibration modesThe pseudo load can come to resemble such loads because it is not derived from physical loadingHaftka, 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

Barthelemys dissertation was mostly about documenting the problem, because there was wide spread skepticism that the problem exists. He was also able to show that the problem is likely to be associated with the pseudo load that you get when you have shape design variables.

In follow-up work, I noted that the stiffness matrix used in finite element calculations have condition numbers that are in the thousands or higher. This means that small errors in the matrix elements or in the loads can lead to large errors in the solution. In fact the condition number is the largest possible amplification factor for the error.

For large amplification of the error, you need the actual loads on the structure to resemble an eigenvector associated with high eigenvalue. That is usually highly oscillatory, and not encountered in actual applications. However, the pseudo load is not physical, and when shape variable are involved, it can have this property of rapid oscillation.

There are now treatments of the semi-analytical method that avoid this problem, in particular one due to Van keulen and colleagues. (Boer and van Keulen, Refined semi-analytical design sensitivities, Int. J. Solids Struct. 37, 6961-6980, 2000)11Problem semi-analyticalDescribe in 45-55 words the pros and cons of the semi-analytical methodUse 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.

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