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On the evaluation of finite element sensitivities to nodal coordinates.

1. Introduction. This short paper is concerned with the evaluation of derivatives of finite element (FE) discretizations with respect to the positions of the nodes in the underlying FE mesh. This problem may arise in a number of different contexts, however, the primary motivation for our work comes from determining the sensitivity of a finite element solution, or some quantity derived from it, to the location of the mesh points. The need for such information arises naturally in the context of design optimization, where a domain shape is sought that yields the optimal performance of some functional of the solution to a partial differential equation (PDE), or system, defined on that domain, [12, 24]. Typical examples of this occur for fluid flow past a body, where one may wish to minimize the drag on the body or maximize the lift or down-force subject to given constraints, [9, 10, 11, 21]. As the shape of the domain is altered smoothly, the FE mesh covering the domain may also be smoothly perturbed and the sensitivity of the solution, or a derived quantity of interest, may be computed. If the solution is obtained via the finite element method, then it is clear, from the chain rule, that the derivative of various quantities with respect to the node positions are required.

Further details of this motivation are presented in Section 2 below. It should be noted, however, that the need to evaluate derivatives of finite element systems with respect to the node positions can arise in other contexts, too. For example, [15] illustrates how general moving grid methods for time-dependent equations and systems may exploit knowledge of derivatives of finite element basis functions with respect to node positions within their formulation. The literature is awash with such methods; see, e.g., [1, 16, 17, 20, 22]. However they are typically restricted to special cases such as linear elements on triangles, for which the differentiation is relatively straightforward [15]. Another potential application comes in research which seeks to establish locally optimal meshes for FE discretizations of elliptic variational problems; see, e.g., [3, 18, 26, 28]. Such problems may be posed by noting that the derivative of the corresponding energy functional with respect to each node position must be zero; see [3, 14].

Of course there are numerous ways in which derivatives of complex expressions with respect to variables such as node locations may be evaluated. Automatic differentiation (AD) has become a topic of much important research in recent years; see, e.g., [4, 8, 13, 27, 29] and many more. This approach allows large subprograms (typically functions) to be differentiated with respect to their input parameters, yielding new subprograms (for evaluating these derivatives) as their output. This technique is extremely powerful. In particular, for large legacy codes, it presents a favourable option; an overview of available AD software can be found on the internet. (1)

However, there are a number of limitations. Even with AD the differentiation of large programs, such as PDE solvers, is not trivial. Defining the task for the AD software alone requires an inexperienced user to get familiar with the AD software, which may not be his primary interest. Further, the code produced by AD is not necessarily (or generally) as efficient as code that is hand-generated. Modern PDE solvers rely heavily on the utilisation of the problem structure to attain their efficiency. As [27, Section 4] describes, applying AD to a PDE solver in a black-box fashion is likely to produce suboptimal efficiency. It is only possible to get (near) optimal performance for the derivative code, if it uses the problem structure. AD has to be applied at just the right portions of the code, and/or the code should be tuned to give good performance with AD.

Thus, implementation of the discrete adjoint "by hand" remains an important option. Further, if approached in a systematic manner, exploiting the knowledge about the formulae that are implemented by the software, then it is a simpler task than one might think. Besides the of use of hand-calculated expressions for the derivatives, as described in this present work, these formulae may also be useful for theoretical analysis of the derivative information in the AD context.

An alternative to AD is simply to evaluate approximations to derivatives based upon finite differences, [7], or related schemes, [19]. This approach is of course easy to implement, simply requiring multiple calls to existing subprograms. However the computational expense when there are large numbers of independent variables can be prohibitive.

A very popular technique to overcome this problem associated with high dimensionality is the discrete adjoint approach [5, 6, 23]. This is introduced briefly in the following section for completeness. Using this technique the entire gradient of a performance functional may be evaluated at a cost comparable to just one additional evaluation of the functional itself. This is in contrast to finite difference approaches, which have a cost equivalent to O(m) functional evaluations, where m is the number of independent variables. In cases where we wish to compute a derivative with respect to the position of each node in a finite element mesh m is typically very large.

Having provided motivation for this work in Section 2, in Section 3 we present the main results of this paper, which provide a derivation of the derivatives of general finite element systems with respect to their node locations. Finally, in Section 4 we present a validation of the results and a discussion of their significance.

2. The Discrete Adjoint Method. Consider a finite element discretization of an elliptic PDE (say) on a domain whose geometry is uniquely defined by a set of parameters, F. The FE solution is defined implicitly by the (possibly nonlinear) algebraic system

(2.1) R(u(F),F) = 0,

where u is the vector of finite element coefficients and R(*, *) = 0 represents the FE equations on the domain defined by F. Let [??](u(F),F) be a scalar-valued function, which depends upon u (F). Since u (F) satisfies (2.1), we may regard this functional as depending only upon F, that is:

(2.2) I (F) = I (u(F), F),

so long as (2.1) holds.

Suppose now that we wish to investigate the sensitivity of the quantity I to perturbations in the parameters F. We begin by considering the effect of small perturbations, [delta]F, of F in (2.2) and (2.1). Discarding higher order derivative terms, such a perturbation results in a perturbation [delta]I in I,

(2.3) [delta]I = [delta][??]/[delta]u [delta]u + [delta][??]/[delta]F [delta]F,

where [delta]u is defined by the perturbation of (2.1) which has to be zero, i.e.,

(2.4) 0 = [delta]R = [delta]R/[delta]u [delta]u + [delta]R/[delta]F [delta]F.

As [delta]R = 0, we can multiply it by an arbitrary term [[psi].sup.T] and subtract it from the right-hand side of (2.3), giving

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This implies that [delta]I may be evaluated without calculating [delta]u provided

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and if u(F) is well-defined by (2.1), then (2.6) uniquely defines [psi], which is of the same dimension as u. (2.6) is known as the adjoint equation and [psi] as the adjoint solution. With this choice of [psi] the perturbation [delta]I is

[delta]I = ([delta][??]/[delta]F - [[psi].sup.T] [delta]R/[delta]F) [delta]F,

revealing the representation of the total derivative

(2.7) DI/DF = [delta][??]/[delta]F - [[psi].sup.T] [delta]R/[delta]F.

The importance of this representation is that, once the original equation (2.1) is solved and I(F) evaluated from (2.2), DI/DF may be evaluated for little more than the cost of a single solve of the linear system (2.6) and a single matrix-vector product in (2.7), regardless of the dimension of F. This is to be compared to other methods for evaluating DI/DF, which typically require the solution of (2.1) (or a linearised version) for each component of F. Evaluating the derivative by means of (2.6) and (2.7) is called the discrete adjoint method; see [6] and the references therein for an introduction.

In the above derivation the specific definition of the parameter vector F is unimportant, and may be anything from coefficients of the PDE, to boundary data, or indeed the shape of the domain.

Now consider the case for which the dependence of the finite element solution upon F is via the FE mesh, as it is for example the case when F defines the shape of the domain. Suppose that the connectivity of this mesh remains constant but that as F is perturbed, the coordinates of the node points in the mesh are perturbed too, as a continuously differentiable function of F. Let the vector s denote the nodal coordinates of each mesh vertex, then we may express s = s(F). Furthermore

(2.8) [delta]R/[delta]F = [delta]R/[delta]s * [delta]s/[delta]F and [delta][??]/[delta]F = [delta][??]/[delta]s * [delta]s/[delta]F

Hence, assuming that the dependence of s upon F is known, or can be determined, the problem of evaluating DI/DF in (2.7) reduces to that of differentiating R and I with respect to the nodal locations s. This is the problem that is addressed in the following section.

One of the simplest alternative approaches to compute DI/DF that one may think of is to use finite differences, i.e., compute u and I for small perturbations of F and use the differences to approximate DI/DF. This approach requires minimal work to program, but the number of times that (2.1) has to be solved is proportional to the dimension of F. Thus if the number of parameters is large this approach is inefficient compared to the discrete adjoint method. The same applies to approaches which solve equations similar to the sensitivity equation (2.4) in order to compute [delta]u/[delta]F and then compute DI/DF by the chain rule. It should be noted however, that if one is interested in the derivatives of not only one dependent variable I but rather a vector of dependent variables, then the sensitivity equation approach is more efficient, if the number of dependent variables I is greater than the number of independent variables F.

In order to see the link to AD, one should note that in AD one generally distinguishes two modes of operation, forward mode and reverse mode. Forward mode essentially applies the chain rule of differentiation to generate [delta]u/[delta]F as an intermediate result and therefore suffers from the same drawbacks as described above. However, reverse mode uses ideas similar to the discrete adjoint and thus (at least in theory) allows to evaluate DI/DF with a cost independent of the dimension of F, with very little programming requirements. However, as already discussed in Section 1, some consideration on how to apply AD with optimal performance may still be necessary; see, e.g., [27].

3. Derivatives with respect to node positions. Equation (2.8) gives rise to the need to compute the Jacobian matrix [delta]R/[delta]s, i.e., the matrix of partial derivatives of the discrete residual vector R(u, s) with respect to the node positions s is required. Let R(u, s) = 0 denote the finite element discretisation under consideration. The Jacobian matrix [delta]R/[delta]s can be computed by summing up element contributions, in a similar way as the stiffness matrix is assembled.

REMARK 3. 1. For the the discrete adjoint technique it is not necessary to compute and store the whole Jacobian, because this Jacobian is only used for a single matrix-vector product [[psi].sup.T] ([delta]R/[delta]s) in (2.7). Thus, it is beneficial not to assemble the whole Jacobian, but to use the local contributions to compute the local matrix-vector products and assemble the resulting vector.

The canonical approach to computing [delta]R/[delta]s is to consider the formulae by which the residual is computed, and to differentiate these. For finite element discretisations of second order elliptic PDE operators, the chain rule of differentiation can be used to reduce this to only two essential terms:

* the derivative of the gradients of the basis functions with respect to the node positions, and

* the derivative of the determinant of the element mapping with respect to the node positions.

In the following this is demonstrated in detail for the Poisson equation as model problem and explicit formulae are given for these two terms. Extension to other PDEs, for example the Lame equation of linear elasticity or the Navier-Stokes equations, is straight forward.

REMARK 3.2. Of course it is also possible to use finite differences or automatic differentiation to compute [delta]R/[delta]s. However, these should be applied on a local level only in order to avoid an increase in computational complexity; see, e.g., [27].

REMARK 3.3. The derivative of the performance criterion [delta][??]/[delta]s can be computed in the same way, if the performance functional is also an integral over a function of the solution and its spatial derivatives, which is usually the case.

Consider the Poisson equation with constant forcing term f - 1. This gives rise to the discrete formulation

(3.1a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.1b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.1c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[phi].sub.i] [member of] [h.sup.1.sub.0] ([OMEGA]) : i = 1,...,N} form a basis of the finite element function space. The integrals contained in the definition of a(*, *) and b(*) are usually computed using numerical integration on each element and then summing over the elements. Hence, the i-th row of the discrete residual vector R may be expressed as

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the outermost sum is over those elements T [member of] [T.sub.h] which have node i as a vertex, n is the number of Lagrange basis functions per element, [[??].sub.l] (f = 1, . . . , m) are the quadrature points on the master element, and [w.sub.l] are the quadrature weights. Further, 9T (j) is the global node number of vertex j of element T, [[nabla].sub.T], denotes the gradient restricted to element T evaluated at the point [x.sub.l] (which corresponds to [[??].sub.l] via the subparametric or isoparametric element mapping), ([??].sup.(j)] ([??]) is the basis function corresponding to node j evaluated at point x within the master element, and [J.sub.T] is the Jacobian of the element mapping. In this expression the parts that are non-constant with respect to the node positions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and [absolute value of det([J.sub.T])],

in each element T. Applying the product rule of differentiation gives for the derivative of [[R].sub.i] with respect to [[[s.sub.k]].sub.t], the t-th coordinate of node k,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Assuming that each of the individual terms in (3.3) can be evaluated, it is now a straight-forward task to compute [delta]R/[delta]s (or [[psi].sup.T][delta]R/[delta]s).

It remains to derive expressions for the derivatives of the gradient [[nabla].sub.T] [[phi].sup.(i)] and the determinant [absolute value of det([J.sub.T])]. The following propostion presents these results.

PROPOSITION 3.4.

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. Let us recall some properties of subparametric (2) finite elements (see, e.g., [2]): the definition of the element mapping function [M.sub.T] (*) (3.6) by means of the shape functions on the master element [??], the resulting expression for the Jacobian [J.sub.T] of the element map ping (3.7), the definition of the ansatz functions [[phi].sup.(i)] on the world element by means of the ansatz functions [[??].sup.(i)] on the reference element (3.8), and the resulting expression for the world element gradient [[nabla].sub.T], [[phi].sup.(i)] (x) (3.9):

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(3.8) [[phi].sup.(i)] (x) : [[??].sup.(i)] ([M.sup.-1.sub.T] (x)),

(3.9) [[nabla].sub.T], [[phi].sup.(i)] (x) = [J.sup.-T.sub.T] [??] [[??].sup.(i)] ([M.sup.-1.sub.T](x)).

By analogy with (3.8) and (3.9) we define

(3.10) [[psi].sup.(i)] (x) : = [[??].sup.(i)] ([M.sup.-1.sub.T](x)),

(3.11) [[nabla].sub.R] [[??].sup.(i)](x) = [J.sup.-T.sub.T] [??] [[??].sup.(i)]([M.sup.-1.sub.T](x)).

Here Sk denotes the coordinate vector of the k-th node of the element T and [??] denotes the gradient on the reference element, i.e., the partial derivatives with respect to the reference element coordinates [??]. Two immediate consequences are

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[delta].sub.i,j] denotes the Kronecker delta.

Due to the application of the numerical integration based on the reference element, the world gradient is only evaluated for points corresponding to the quadrature points on the reference element [[??].sub.l],

(3.13) [[nabla].sub.T][[phi].sup.(i)] : = [[nabla].sub.T] [[phi].sup.(i)]([M.sub.T] ([[??].sub.l])) = [J.sup.-T.sub.T] [??][[phi].sup.(i)] ([[??].sub.l]).

Since [??] [[??].sup.(i)] ([[??].sub.l]) is independent of the node positions, [[nabla].sub.T] [[phi].sup.(i)] depends on the node positions only via the transpose of the element Jacobian [J.sup.T.sub.T]. The derivatives with respect to the node positions can be calculated using the implicit function theorem on the reformulated (3.13),

(3.14) 0 = [J.sup.T.sub.T] ([[[s.sub.k]].sub.t])[[nabla].sub.T] [[phi].sub.(i)] - [??][[??].sub.(i)] ([[??].sub.l]).

This gives

(3.15) [delta[[[nabla].sub.T][[phi].sup.(i)]/[delta][[[s.sub.k]].sub.t] = -[J.sup.-T.sub.T] [delta][J.sup.T.sub.T]/[[[s.sub.k]].sub.t] [[nabla].sub.T][[phi].sup.(i)],

where the derivative of the transposed Jacobian is in the component wise sense. Taking a closer look at the derivative of the u-th component of [[nabla].sub.T] [[phi].sub.(i)], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (by (3.12))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (by (3.11))

This shows the first part of the proposition.

The derivative of [absolute value of det([J.sub.T])] also can be reduced to a derivative of [J.sub.T] due to the identity

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which follows from expansion to sub-determinants and the adjoint representation of the inverse of the matrix A. Applying this formula and an analogue of (3.12), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

4. Numerical verification. In [25, Section 2.7.2.2], for verification purposes, derivative values obtained by the discrete adjoint method, using the expressions derived in Section 3, have been compared to those obtained by the finite difference approximation

(4.1) DI/D[F.sub.i] [approximately equal to] I ([F.sub.i] + h) - I ([F.sub.i] - h)/2h

(CD, central difference), with h = [10.sup.-5]. The examples used in [25] use Taylor-Hood ([P.sub.2]/[P.sub.1]) finite element discretisations of the incompressible Navier-Stokes equations. Here we give results obtained for Example 2.2 from [25], because these are representative for many more tests which have been undertaken, and because they highlight the advantage of the discrete adjoint method compared to finite differences.

[FIGURE 4.1 OMITTED]

[FIGURE 4.2 OMITTED]

The example comprises of an obstacle moving in a channel of viscous fluid at steady state at Reynolds number Re = 20. Under consideration are the derivatives of the drag with respect to shape parameters defined by Bezier splines, resulting in ten parameters. See Figure 4.1 for the domain geometry, and Figure 4.2 for an illustration of the shape parameterisation.

As the number of shape parameters is very moderate in comparison to the number of mesh points in this example, we follow the suggestion of an anonymous referee to include a further alternative, where in the adjoint representation (2.7) the derivatives [delta]I/[delta]F and [[psi].sup.T]([delta]R/[delta]F) are evaluated by central differences. This allows to better judge the cost of computing those derivatives by the methodology from Section 3 in comparison to the overhead of the mesh deformation. Values computed by this approach will be denoted by CDR.

Table 4.1 lists the derivative values as computed by the discrete adjoint method, central difference (CD), the approach described above (CDR), and the relative difference between the adjoint, and central difference values. The tests have been performed on a Linux desktop PC with Intel(R) Core(TM)2 CPU 6400 at 2.13GHz, using the single threaded C code FEINS of the first author, using meshes producing the number of degrees of freedom as listed in the table (#DOFs). The derivative values compare well and the most significant relative errors occur only for those components of the gradient with small absolute value. In those cases the relative error of the finite difference approximation is largest due to cancellation effects. Note that smaller values of h (<< [10.sup.-5]) in (4.1) led to unreliable results for the finite difference approximation due to cancellation effects and the inexact solves of the discretised Navier-Stokes systems. The differences between the adjoint and CDR values are significantly smaller, as they use the same (inexact) solution of the adjoint equation (2.6) and avoid the differentiation of the inexact solution of the discrete Navier-Stokes system.

Table 4.1 also shows the total time taken for the computation of the value of I ([t.sub.solve]) and those taken for a subsequent derivative computation by each method ([t.sub.adj], [t.sub.CD], and [t.sub.CDR], respectively). Note the different exponent in [t.sub.CD]. The computation by the central differences already takes advantage of the small perturbations in the nonlinear system, initialising the perturbed nonlinear solves with the unperturbed solution, thus resulting in [t.sub.CD] [approximately equal to] 5.9 * [t.sub.solve] rather than the 20 * [t.sub.solve] one might expect. As the discrete adjoint technique requires only one solution of the linear adjoint equation (2.6), rather than multiple solutions of linearised equations in Newton's method for solving the nonlinear Navier-Stokes system, it can be observed that [t.sub.adj] [approximately equal to] 0.67 * [t.sub.solve], even less than [t.sub.solve]. Overall, the advantage of the adjoint method is evident.

REMARK 4. 1. A comparison with automatic differentiation is omitted here, but may be a subject of future work.

For more details on the example problem and its implementation, as well as further examples, see [25].

5. Conclusions. A systematic approach to differentiation of finite element discretisations with respect to the coordinates of the nodes in the FE mesh has been presented. The problem can be reduced to the derivatives of only two terms depending on the mesh geometry, for which expressions have been presented in Section 3. Results obtained with the presented approach have been verified by comparison to those obtained with finite difference approximation for an example problem. The example also demonstrates the efficiency of the methods discussed in this work.

Applications of the result range from shape optimisation to moving mesh FE discretisations, and possibly many more.

* Received October 12, 2007. Accepted for publication May 5, 2008. Published online on March 4, 2009. Recommended by T. Apel.

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(1) See http://www.autodiff.orq.

(2) The clasical Lagrange finite elements belong to this class of elements, for isoparametric elements one simply subsitutes [psi] = [phi], [??] = [??].

RENE SCHNEIDER ([dagger]) AND PETER K. JIMACK ([double dagger])

([dagger]) TU Chemnitz, Fakultdt fdr Mathemetik, 09107 Chemnitz, Germany (rene.schneider@mathematik.tu-chemnitz.de).

([double dagger]) University of Leeds, School of Computing, Leeds LS2 9JT, UK (pkj@comp.leeds.ac.uk).
TABLE 4.1
Verification of adjoint derivative evaluation in the EE discretisation
by comparison to finite difference values.

adjoint CD CDR (CD-adj)/[absolute
 value of CD]

1.799e-01 1.799e-01 1.799e-01 3.9e-05
4.912e-01 4.912e-01 4.912e-01 1.3e-05
1.118e+00 1.118e+00 1.118e+00 3.8e-06
7.675e-01 7.675e-01 7.675e-01 2.7e-06
-6.314e-02 -6.311e-02 -6.314e-02 -4.9e-04
-3.683e-02 -3.677e-02 -3.683e-02 -1.4e-03
-5.934e-02 -5.934e-02 -5.934e-02 -4.6e-05
-1.702e-01 -1.702e-01 -1.702e-01 -2.5e-05
2.665e-01 2.665e-01 2.665e-01 2.9e-05
1.644e-02 1.644e-02 1.644e-02 -1.8e-05

Re = 20 (v = 0.05), #, DOFs = 572608, [t.sub.solve] = 3.9e+03s
[t.sub.adi] = 2.6e+03s [t.sub.CD] = 2.3e+04s tCDR = 3.6e+03s
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