Theoretical aspects and applications of goal-oriented reanalysis methods

1 INTRODUCTION

In many applications, design modifications (design changes) are investigated. The design variables are, for instance, parameters that describe the cross-sections, the geometry, the topology or material parameters, and so forth. Classical applications with successive design steps are, for example, structural optimization, reliability analysis, or structural damage analysis. Hundreds or even thousands of different design configurations are investigated in many cases, and the state equation must be solved for each design step. The repeated structural analysis of large problems involves significant computational effort.

An efficient reanalysis method can be used to reduce the overall computational cost. The goal of classical global reanalysis procedures is to evaluate the change in the state variables for some given design modifications efficiently and with sufficient accuracy without directly solving the set of modified equations of the changed problem. Reanalysis techniques for the computation of changes in the state variables due to given design changes have been extensively studied in the literature, for example [1-6], among many others. It has been formulated for static and dynamic problems. Furthermore, reanalysis methods have been applied to sensitivity analysis and optimization problems [7-9].

Very often, we are interested only in certain quantities of interest J, and so-called goal-oriented or duality techniques or adjoint state methods can be used to compute J. This is known as the concept of influence functions in structural mechanics, see for example [10]. The quantities of interest could be point values, for example, a displacement component, a stress component at a point, or some integral value.

This contribution presents a method for predicting the change in a quantity of interest due to structural modifications using the primal and dual solutions changes. The changes in the primal and dual solutions are computed by using the residual increment approximation (RIA) method [6]. This reanalysis procedure is based on a formulation in terms of residual increments. In contrast to other existing reanalysis methods (e.g., the CA method [11]), which are based on the evaluation of changed stiffness matrices, only residual vectors must be computed and stored. Hence, this yields an efficient goal-oriented reanalysis method to compute the change in the quantities of interest due to given design modifications with sufficient accuracy without directly solving the set of modified equations of the changed problem.

2 FORMULATION OF GOAL-ORIENTED ANALYSIS

2.1 The quantities of interest

We are interested in a quantity of interest J in goal-oriented or duality techniques. We consider a quantity of interest which depends on the state variables and on some design variables , that is, parameters which describe the cross-sections, the geometry, the topology or material parameter, and so forth. In this paper, the quantity is linear in , but possibly nonlinear in . Furthermore, we suppose that J is differentiable with respect to .

The quantities of interest could be point values, for example, a displacement component or a stress component at a point . Furthermore, it can be some integral value, for example,

(1)

2.2 The primal and dual (adjoint) problems

The state is determined by the primal problem in terms of a residual . This paper considers linear problems, that is, is linear in . For a given fixed design , the primal solution is given from

(2)

where is the stiffness matrix and is the primal load vector. Both, and depend only on .

For a chosen , the corresponding dual or adjoint solution, influence function or generalized Green's function is determined by the so-called dual problem written in terms of the dual residual vector . For a given fixed design , the dual solution is given from

(3)

Here, is the so-called dual load vector. For self-adjoint problems, we have .

It is important to note that the residual vectors and and the quantities of interest J are linear in and , respectively, but possibly nonlinear in .

2.3 Computing the quantity of interest

The quantity of interest J can be computed in two different ways. Classically, the primal problem is solved and can be computed in a post-processing step.

Alternatively, the dual solution can be used, which can have a significant advantage.

With the primal problem (2) and the dual problem (3), we have and respectively, and hence

(4)

Finally, for a given fixed design , the quantity J is given by evaluating a scalar product, that is,

(5)

or alternatively from

(6)

The big advantage of the formulation (6) can be summarized as follows: If the dual solution is known, the quantity of interest can be computed for arbitrary primal load vectors just by a simple scalar product of and . This is, for instance, used in structural mechanics in the context of influence functions to compute J for many different load cases.

3 GOAL-ORIENTED REANALYSIS

Reanalysis is used in many fields which are concerned with design modifications. Let be a given initial design and let be the corresponding solution of (2). Assume a changed design and let be the corresponding solution.

Furthermore, let be the value of J for the initial design and let be the value of J for a given changed design .

The general goal-oriented reanalysis problem can be stated as follows: Find the change

(7)

of J due to given design changes with sufficient accuracy without solving the complete modified equations.

3.1 The initial and changed primal and dual problems

The primal and dual problems for the initial and changed designs according to (2) and (3) are given as

(8)

(9)

and

(10)

(11)

where , , , , and .

3.2 The change in the quantity of interest

The change in the quantity of interest can be computed in two different ways. Firstly, we use a formulation with and the change . With (5) we obtain after some manipulations

(12)

The change depends on and the change .

Alternatively, the change in J can be expressed in terms of and the change . By using (6) we obtain

(13)

In this formulation, the change depends on and the change .

Finally, according to (5) and (6), we obtain the two equivalent relations (12) and (13) to express the change .

The big advantage of the second formulation (13) can be summarized as follows: If the dual solution and the change for a fixed are known, the change in the quantity of interest can be computed for arbitrary primal load vectors and changes , that is, can be computed for many different primal load cases just by evaluating simple scalar products.

4 COMPUTING THE CHANGE IN THE QUANTITY OF INTEREST

To compute , we have to evaluate (12) or (13). The changes in the primal and dual loads and can easily be computed with less computational effort. The only challenge is the computation of approximations of the increments or with sufficient accuracy. This can be done by using reanalysis methods. A general efficient reanalysis method for the computation of an approximation of based on residual increment approximations (RIA method) has been presented in [6]. The paper uses this method to compute the changes and with sufficient accuracy without solving the complete modified equations.

4.1 Reanalysis for the primal problem

The starting point for the reanalysis method is the residual of the changed problem defined in (10). Although, the problem is linear in , the residual is in the general case nonlinear in , that is,

(14)

The changed problem (10) can be expressed in terms of the initial design as

(15)

Hence, the residual increment with respect to is given as . Furthermore, we obtain with for the first term on the right side in (15) the relation

(16)

Finally, Equation (15) leads to

(17)

Note that the residuum is not zero for some approximation of . The above equation can be expressed as the recurrence relation

(18)

Explicitly, we set the initial value . Then, recurrence yields the first two values from

with

For all other values we have

(19)

After n iterations, the state approximation for the changed design is obtained as .

The stiffness matrix is the same as used for the solution of the initial design and, therefore, is usually already given in the decomposed form. Therefore, the computation of in (18) requires just forward and backward substitution. Only residual vectors must be computed and stored within the reanalysis method.

The reanalysis procedure adapted from Equation 18 is a local approximation based on information calculated at a single point (). The results can be improved using a vector-valued rational approximation method introduced in [12] and applied to linear reanalysis problems in [4].

This method is used within the numerical examples in the present paper. Details about this method and the overall algorithm of the reanalysis method are given in [6].

4.2 Reanalysis for the dual problem

In the same way, as for the primal problem, we can formulate a reanalysis method for the dual problem, that is, we want to compute an approximation of to evaluate the relation (13).

The changed problem (11) can be expressed in terms of the initial design as

(20)

Hence, the residual increment with respect to is given as . Furthermore, we obtain with for the first term on the right side in (20) the relation

(21)

Finally, Equation (20) leads to

(22)

Note that the residuum is not zero for some approximation of . Finally, this can be expressed as the recurrence relation in the form of

(23)

Explicitly, we set the initial value . Then, recurrence yields the first two values from

with

For all other values we have

(24)

After n iterations, the approximation of the dual solution for the changed design is obtained as .

In the same way, as for the primal problem, the results are improved by using a vector-valued rational approximation method. This is used within the numerical examples. Details and the overall algorithm are given in [6].

4.3 First-order adjoint sensitivity relation

In many applications, the classical first-order approximation (FOA) is used to predict the changes in the state variables or quantities of interest due to design modifications, see for example [13, 14]. The results are valid only for very small design changes. In this study, we compare the proposed reanalysis method with the classical FOA for completeness.

To compute the changes in a quantity of interest, so-called adjoint sensitivity analysis can be used, see, for example [15]. For a given fixed design change we obtain the first-order approximation for the change in the quantity of interest in the form of

(25)

The matrix is the co-called pseudo load matrix, see for example [13, 14] for details and explicit formulations.

The advantage of the first-order approximation (25) is that the relation depends only on the known initial primal and dual solutions and , that is, the changes in the primal and dual solutions are not required. The disadvantage of this method is that it requires the derivatives with respect to the design variables, that is, we have to compute and . This can be very difficult and expensive in many situations and model problems. However, it is a robust method and is used in many applications. Therefore, we compare the FOA with the other proposed methods within the numerical examples.

5 NUMERICAL EXAMPLE

In the above sections, we have considered three methods to compute an approximation of the change for a given fixed design change . The methods are summarized in Table 1 and investigated in the following numerical example. We consider the model problem of linear elasticity.

TABLE 1. Summary of different methods to compute approximations of the change .

Method	Discrete formulation
1		(see Equation 12)
2		(see Equation 13)
3		(see Equation 25)

Approximations of the changes in the primal and dual solutions and required in methods 1 and 2 are computed by using the reanalysis method from (18) and (23), respectively. Furthermore, the accuracy of and are improved using a vector-valued rational approximation method as described in [6].

We examine a bi-material solid under tension with large deformations; see Figure 1A. The body is clamped on the left side and loaded by traction . The design variables are the Young's moduli E₁ and E₂ in the two domains, that is, and . For simplicity, we consider only changes in the second variable E₂ and keep E₁ constant. For the initial design , we chose . The Poisson's ratio for both domains is , and the finite element discretization consists of classical bilinear Q4 elements.

The quantity of interest is the stress component at point , that is, . The dual load case (red arrows in Figure 1D) causes an approximation of a unit dislocation at point . The primal and dual solutions for the initial design are given in Figures 1B and E.

In this example, a significant change in Young's moduli is considered, that is, we investigate a design change . This yields a significant change in the primal solution; see Figure 1C. The difference in the dual solution due to the design change is relatively small; see Figure 1F.

The values of J for the initial and changed designs, as well as the exact change, are given as

(26)

Approximations of the changes in the quantity of interest are computed using the three methods given in Table 1. The results are stated in Table 2.

TABLE 2. Bi-material solid under tension (): Accuracy of the different methods (see Table 1).

Results for :			Results for :
Method	Approximation of	Rel. error [%]	Method	Approximation of	Rel. error [%]
1	2.1349	3.7241	1	2.1878	1.3397
2	2.1619	2.5076	2	2.2175	0.0018
3	2.7415	23.628	3	2.7415	23.628

Results for :			Results for :
Method	Approximation of	Rel. error [%]	Method	Approximation of	Rel. error [%]
1	2.2179	0.0164	1	2.2175	0.0014
2	2.2176	0.0027	2	2.2175	0.0001
3	2.7415	23.628	3	2.7415	23.628

• Note: The results are given for different n, which is the number of iterations used within the reanalysis in methods 1 and 2. The relative errors are given w.r.t. the exact change .

Method 1 and 2 yield very accurate results even for significant design changes. Method 3 is just a first-order relation and yields for significant design changes only a rough approximation. Furthermore, method 2 outperforms method 1 in this example because the change in the dual solution due to design changes is relatively small. Hence, we can obtain a good approximation of with few iterations within the reanalysis method. In contrast, the change due to design changes is huge. Therefore, we need more iterations within the reanalysis method to compute with high accuracy.

6 CONCLUSIONS

Reanalysis methods are very useful in reducing the computational effort within applications concerned with multiple design modifications, such as structural optimization, reliability analysis, or damage analysis. In this contribution, a goal-oriented reanalysis method for the prediction of the change in a quantity of interest due to structural modifications by using the changes in the primal and dual solutions has been presented.

The changes in the primal and dual solutions due to given design modifications are computed using the RIA method. This reanalysis procedure is based on a formulation in terms of residual increments. In contrast to other existing reanalysis methods (e.g., the CA method), which are based on evaluating changed stiffness matrices, only residual vectors must be computed and stored. The proposed method is very general and can be used for different design modifications. The overall goal-oriented reanalysis procedure is straightforward and can easily be implemented in existing finite element programs because no derivatives with respect to the design variables are necessary. The numerical example demonstrates that the method yields accurate results even for significant design changes. In the present paper, only linear problems are considered. The extension to nonlinear problems will be discussed in future work.