# Sensitivity Analysis in Structural and Multidisciplinary Problems

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_265-1

- 81 Downloads

## Synonyms

## Definitions

This entry is an introduction to sensitivity analysis and some applications in different topics related to continuum mechanics. Two main approaches, the direct method and the adjoint system one, are presented for both discrete and continuous design parameters. The theoretical investigation is supported by some illustrative examples in order to make more clear the analysis. Some notes on second-order sensitivity analysis are also included.

## Introduction

Many continuous systems are characterized by parameters, the values of which deeply affect their resulting mechanical behaviors under different conditions. Therefore, in a smart design procedure, it is very important to fix these values in order to fit a certain desired behavior or at least to investigate what are the effects connected to these modifications. Such a study is called sensitivity analysis: One is considering some objective functions which refer to selected properties of the system under investigation, and sensitivity analysis provides information about their dependence on the design parameters. The aim of this entry is to show how to extract this information: Firstly the focus will be on continuous design parameters bringing the so-called variational sensitivity analysis, and then the discussion will move to discrete sensitivity analysis, which is a very general case since finite element analysis always provides discrete systems described in terms of stiffness matrices and nodal vectors. In both situations two approaches will be discussed: the direct method and the adjoint system one. The choice between the two usually depends on the ratio of the number of design parameters to the number of objective functions, as it will be shown in the next sections. In order to make more concrete the theoretical discussion, an example coming from structural mechanics will be presented and studied in detail. However, already from these few lines, it is clear that sensitivity analysis can find interesting applications in any field, especially when one is interested in realizing a certain production process and reducing the associated costs. It plays also an important role in formulation of optimization problems; see, e.g., Lekszycki (2018). The main part of the text will be dedicated to the so-called first-order sensitivity analysis. However, a short digression on second-order sensitivity analysis will also be developed: such a study can be useful when more accuracy is required or when the effect of small changes of optimal values is investigated. For the sake of compactness, a selection of the topics related to sensitivity analysis has been performed, and for more details, the interested reader can refer, for instance, to the books (Haftka and Gürdal, 2012; Choi and Kim, 2006a), which have been the main sources of this entry.

## Variational Sensitivity Analysis

*Ω*

_{1}is the part of the boundary where the displacement field, \(\mathbf {u}\left ( \mathbf {X} \right ) = \chi \left ( \mathbf {X} \right )-\mathbf {X} \), is prescribed, whereas

*Ω*

_{2}is the region of the boundary where the external traction is assigned, and

*Ω*

_{0}=

*Ω*

_{1}+

*Ω*

_{2}. Here

**X**denotes the coordinate of a material particle in the reference configuration \(\mathcal {B}_0\), and

**n**

_{0}is the normal vector to the surface

*Ω*

_{0}. In order to explain how the procedure progresses, the relationship between the Green strain tensor E and the displacement vector

**u**is linearized, that is, \(\mathsf {E}=\frac {1}{2}\left ( \left ( \nabla \mathbf {u}\right )^T+\nabla \mathbf {u} \right )\), so that the previous equation can be written according to the principle of virtual work as follows:

*L*, whose action on the displacement field

**u**will be denoted by

*L*(

**u**). Therefore, one could replace Eq. 1 by the expression:

*δ*

**u**, which is any kinematically admissible variation of the field

**u**, and integrating over the domain \(\mathcal {B}_0\), one obtains the following chain of results:

*L*

^{a}represents the adjoint differential operator associated with

*L*and the chosen scalar product. The linear operators

*B*and

*B*

^{a}are defined on the boundary of the body \(\mathcal {B}_0\), and they have been obtained after integration by parts. A closer look at the boundary term permits to write it as follows:

*δ*

**u**along the surface

*Ω*

_{2}.

*K*

_{j}, which describe some properties of the body under investigation, with respect to design variables, which in this section will be described as continuous fields

*b*

_{p}(⋅) (the terms design variables or parameters refer to all the variables which characterize the configurations of the continuous systems and can be modified during the process). From a mathematical point of view, this dependence is quantitatively expressed by means of the first Gateaux derivative of the functionals

*K*

_{j}with respect to design parameters, i.e.:

*K*, will be analyzed in the rest of the entry. In particular this functional is written as the sum of four terms:

*b*

_{p}(⋅) is both explicit, in

*g*

_{0}and

*g*

_{1}, and implicit through the displacement and the stress fields. In order to compute the sensitivity of this functional, there are two approaches which can be adopted: the direct method and the adjoint system one.

### Direct Method for Sensitivity Analysis

*g*

_{0},

*g*

_{1},

*g*

_{2},

*g*

_{3}are all differentiable with respect to their arguments, the first derivative of the functional (7) can be expressed as follows:

By solving these equations, the final expression for the first-order sensitivity of the objective functional *K* can be eventually computed. However, despite its simplicity, this method requires the solutions of *P* additional equations, where *P* is the number of design variables. Consequently its cost increases as this number grows up.

### Adjoint System Method for Sensitivity Analysis

The main idea underlying the adjoint system method for the computation of sensitivity of objective functionals consists in expressing the variation of the considered functionals in terms of two families of fields, one referring to a primary continuous system and the other to a fictitious adjoint system. The additional fields for the adjoint system are introduced as Lagrange multipliers (see dell’Isola and Di Cosmo 2018 for an introduction to this method), in an extended objective functional which will include also balance equations and constitutive relations as constraints. In order to make a more direct comparison with the method outlined in the previous subsection, the main focus of the following discussion will be on a continuous body described as a linear elastic material, obeying the equations earlier introduced (see Eqs. 1, 2, 3, and 11). In addition to the aforementioned sources, the interested reader can refer also to Dems and Mroz (1983) and Tortorelli and Michaleris (1994).

*H*be the following extended functional:

^{a}are the fields referring to the adjoint structure and they are independent of the fields without this index, which refer to the primary structure. Therefore, the variation of this functional with respect to design variables gives the following result:

## Discrete Sensitivity Analysis

Many problems in continuum mechanics require the introduction of discretizations and the use of numerical approximation techniques in order to find a solution. If before the recent advances in computer technology these methods were rarely used and other scheme of approximation were widely spread, nowadays numerical methods and discretized systems are strictly connected to the world of continuum mechanics, and computational mechanics is an extremely active research field (see, for instance, Kaessmair and Steinmann 2018; Turco et al. 2016). Finite element methods (see de Borst 2018), in particular, have become a tool which is implemented in all the softwares which are used for solving mechanical problems. Therefore, discrete sensitivity analysis involves many nontrivial situations, being the proper approach after discretization techniques have been applied. Furthermore discrete sensitivity analysis can be helpful to understand how the change of some parameters in a mathematical model could affect a chosen response functional: Such a study is, actually, fundamental to fit the constitutive parameters of suitable mathematical models. Additional references regarding this topic are, for instance, the papers (Van Keulen et al., 2005; Adelman and Haftka, 1986).

**u**as follows:

**f**is a vector load. Let

*G*(

**u**,

**b**) be the objective function for the considered problem. The sensitivity of

*G*with respect to the design variables

**b**is made up of two terms, the explicit dependence on the variables and the one implicitly contained in the nodal vector

**u**. In formulas one can write:

### Direct Method for Discrete Sensitivity Analysis

*G*:

*G*. In particular, these derivatives could be efficiently computed also by means of finite difference methods (this approach is called

*semi-analytical method*for sensitivity analysis). However, such an approximation suffers from accuracy problems as several authors have illustrated in the literature. These accuracy problems, in particular, becomes more evident in beam or plate theory, where the error in the semi-analytical method increases as the mesh becomes more refined (Cheng and Olhoff, 1993; Pedersen et al., 1989).

### Adjoint Method for Discrete Sensitivity Analysis

The adjoint method for the computation of sensitivity \(\frac {dG}{db_p}\) is based on the introduction of an additional field, which plays the role of the displacement nodal vector of an adjoint system. An equilibrium equation is derived for this adjoint response vector, the solution of which will be used for the evaluation of the sensitivity \(\frac {dG}{db_p}\).

*H*be the extended function:

*λ*is a Lagrange multiplier introduced for the equilibrium constraint. The derivative with respect to the design variables of this extended function will be written as follows:

**u**ca be chosen to satisfy the equilibrium condition in Eq. 18, whereas the Lagrange multiplier can be selected in order to eliminate the coefficient of the sensitivity \(\frac {d\mathbf {u}}{db_p}\), which means:

**u**and

*λ*, which can be interpreted, from a mechanical point of view, as nodal vectors of the primary and the adjoint systems.

### Nonlinear Sensitivity Analysis

*q*is the nonlinear internal force,

**f**is the external load, and

*μ*is a scale parameter which is explicitly used to take into account for the whole loading process, starting at zero load. The objective function remains

*G*(

**u**,

**b**) and the corresponding sensitivity is written in Eq. 19. In order to evaluate the quantity \(\frac {d\mathbf {u}}{db_p}\), one can differentiate Eq. 27 with respect to the design variables, obtaining the following equation:

**u**and the tangent stiffness J can be computed by means of Newton’s iterative methods (see Haftka and Gürdal 2012). Therefore one gets:

*λ*is solution of the adjoint linear problem:

## Example: A Vibrating Beam with an Elastic Foundation

*y*(

*x*,

*t*) is the vertical displacement of the beam. In this example the beam will be a steel beam, so that

*E*and

*ρ*are the Young modulus and the density of steel, respectively. The cross section of the beam is a square with side

*a*

_{c}= 2 cm and area

*A*= 4 cm

^{2}, whereas its length is

*L*= 2 m. \(K(x)= \frac {K_0 x (L-x)}{L^2}\) is the nonuniform stiffness of the spring foundation, with \(K_0=1\mathrm {e}^{6} \frac {\mathrm {N}}{\mathrm {m}^2}\). The amplitude of the load is

*F*

_{0}= 10N, and its frequency is \(\nu = \frac {\omega }{2\pi }=40\) Hz.

*u*(

*x*):

*x*=

*s*in the reference configuration, which is the interval [0,

*L*], and let the design variable be the value of the stiffness

*K*(

*x*) at the point

*x*=

*s*

_{0}.

*G*with respect to the design variable will be computed using the adjoint method, in order to show the mechanical realization of the adjoint system. Therefore, let

*H*be the extended functional:

*v*

^{a}(⋅) will satisfy the same boundary conditions as the primary field (even if, in more general situations, the boundary conditions could be different). By performing the variation with respect to the design variable and imposing the constraint in Eq. 34 one gets that the sensitivity \(\frac {\delta H}{\delta K(s_0)} = \frac {\delta G}{\delta K(s_0)}\) can be written as follows:

*v*

^{a}(⋅) the solution of the equation:

*x*=

*s*, which is also the material point involved in the definition of the objective functional

*G*=

*u*(

*s*).

*ComsolMultiphysics*, which uses standard finite element methods for solving the problem. Both the primary and the adjoint solutions,

*u*(⋅) and

*v*

^{a}(⋅), respectively, have been computed, and the results have been plotted in Figs. 1 and 2. A parametric study has been computed, where both the positions

*s*and

*s*

_{0}have been varied.

In particular the chart plotted in Fig. 1 shows the value of the sensitivity as a function of the position *s*_{0} for different values of the parameter *s* in the objective functional. Instead, the diagram in Fig. 2 presents the maximal and minimal values of the sensitivity over the set of possible *s*_{0} as a function of the position *s*. One can notice that the values of the sensitivity are higher, in absolute value, when *s* is closer to the right end of the beam, whereas a second lower peak can be observed between the left end and the middle point of the beam. From this example, it is also possible to notice that sensitivity analysis is important also from another point of view: It can be used to obtain the direction of maximal growth, providing useful information for optimization algorithms.

## Second-Order Sensitivity Analysis

Under some circumstances, for instance, when one is interested in the sensitivity of an optimal solution, the information coming from the first-order sensitivity analysis is not sufficient, and it is needed the knowledge of second-order derivatives of the objective functional *G*. In this section the discussion will focus on discrete system, the variational approach being an immediate generalization (the interested reader can refer to Dems and Mroz 1985). The two approaches, the direct method and the adjoint one, are still available, but the adjoint method usually requires a less number of solutions to compute.

**u**obeying the equilibrium condition:

*G*(

**u**,

**b**) and its second derivative can be easily computed:

*G*. Despite its simplicity, this method is numerically costly since it grows quadratically in the number of design variables.

*P*solution of the direct method (where

*P*is the number of design variables) and the adjoint field

*λ*. Therefore, this method is more advantageous with respect to the direct one from the point of view of numerical implementation.

## Concluding Remarks

The main aim of this entry is to provide an introduction to the methods of sensitivity analysis for mechanical problems. This introduction, of course, cannot be considered exhaustive since some topics have not been investigated and other ones have been only marginally approached. Therefore, some remarks are required in order to conclude this work.

First of all, even if the entry focused on elasticity, multidisciplinary problems can be treated as well. In particular, many authors extended the methods presented in this entry to coupled systems, like thermoelastic or thermoplastic (see Dems 1987, 1986; Dems and Mroz 1987), or to biological systems (see Coelho et al. 2011), or to thermofluids (see Tortorelli et al. 1991; Smith et al. 1998a). For instance, dealing with solid-acoustic interaction phenomena, it is relevant to study the sensitivity of functionals, like natural frequencies, eigenvectors and amplitudes, or, when the domain is unbounded, energy flux and directivity of the radiation. Indeed, these characteristics of the acoustic vibrations are extremely important in order to study wave propagation or to reduce vibrations of systems (see Christensen et al. 1998a,b).

Another interesting application of sensitivity analysis to multidisciplinary topics is related to the work by Smith et al. In some papers, indeed, sensitivity analysis has been applied to the study of extrusion of melt polymers (see Smith et al. 1998b,c). The motion of the melt polymer is modeled according to Hele-Shaw flow model, and sensitivity analysis is applied to functional like the inlet pressure or the exit velocity, parameters which deeply affect the cost and the quality of the extrusion process.

Even if some topics should need more details, like nonlinear sensitivity analysis and second-order variations, an entire chapter of sensitivity analysis, which is shape sensitivity analysis, has not been discussed, and, consequently, some remarks concerning this field will be added in this concluding section. In shape sensitivity analysis, one is interested in the dependence of objective functionals on the variation of shape of the body which constitutes the system. The main tool that is used to evaluate this variation is the so-called material derivative: Under a shape variation, the particles of the body are transported along a flow, which is parameterized by a time-like parameter *τ*. The total derivative of a physical quantity *ψ* with respect to this parameter, computed considering the point *x* as a material particle moving along the aforementioned flow, is the material derivative \(\frac {d\psi }{d\tau } = \frac {\partial \psi }{\partial \tau } + \frac {dx^j}{d\tau } \frac {\partial \psi }{\partial x^j}\), where \(\frac {dx^j}{d\tau }\) play the role of generalized velocity fields. Replacing the derivative with respect to design variable with the material derivative abovementioned, one can generalize direct and adjoint methods to this more complex situation. Material derivatives of volume and surface elements can be computed, producing additional terms which involve, for instance, the mean curvature of the boundary. Such a method has been widely studied in literature (see Dems and Mroz 1984; Choi and Kim 2006a), and interesting applications to multiphase problems have been also considered.

## Cross-References

## References

- Adelman HM, Haftka RT (1986) Sensitivity analysis of discrete structural systems. AIAA J 24(5):823–832CrossRefGoogle Scholar
- Cardoso J, Arora J (1988) Variational method for design sensitivity analysis in nonlinear structural mechanics. AIAA J 26(5):595–603MathSciNetCrossRefGoogle Scholar
- Cheng G, Olhoff N (1993) Rigid body motion test against error in semi-analytical sensitivity analysis. Comput Struct 46(3):515–527CrossRefGoogle Scholar
- Choi K, Kim N (2006a) Structural sensitivity analysis and optimization 1: linear systems. Springer Science & Business Media, New YorkGoogle Scholar
- Choi K, Kim N (2006b) Structural sensitivity analysis and optimization 2: nonlinear systems and applications. Springer Science & Business Media, New YorkGoogle Scholar
- Christensen S, Sorokin S, Olhoff N (1998a) On analysis and optimization in structural acoustics part I: problem formulation and solution techniques. Struct Optim 16(2–3):83–95Google Scholar
- Christensen S, Sorokin S, Olhoff N (1998b) On analysis and optimization in structural acoustics part II: exemplifications for axisymmetric structures. Struct Optim 16(2–3):96–107Google Scholar
- Coelho P, Fernandes P, Rodrigues H (2011) Multiscale modeling of bone tissue with surface and permeability control. J Biomech 44(2):321–329CrossRefGoogle Scholar
- de Borst R (2018) Finite element methods. Springer, Berlin/Heidelberg, pp 1–8. https://doi.org/10.1007/978-3-662-53605-6_13-1 Google Scholar
- dell’Isola F, Di Cosmo F (2018) Lagrange multipliers in infinite-dimensional systems, Methods of. Springer, Berlin/Heidelberg, pp 1–9. https://doi.org/10.1007/978-3-662-53605-6_185-1 Google Scholar
- dell’Isola F, Seppecher P, Corte AD (2018) Higher gradient theories and their foundations. Springer, Berlin/Heidelberg, pp 1–10. https://doi.org/10.1007/978-3-662-53605-6_151-1 Google Scholar
- Dems K (1986) Sensitivity analysis in thermal problems I: variation of material parameters within a fixed domain. J Therm Stresses 9(4):303–324MathSciNetCrossRefGoogle Scholar
- Dems K (1987) Sensitivity analysis in thermal problems II: structure shape variation. J Therm Stresses 10(1):1–16MathSciNetCrossRefGoogle Scholar
- Dems K, Mroz Z (1983) Variational approach by means of adjoint systems to structural optimization and sensitivity analysis I: variation of material parameters within fixed domain. Int J Solids Struct 19(8):677–692MathSciNetCrossRefGoogle Scholar
- Dems K, Mroz Z (1984) Variational approach by means of adjoint systems to structural optimization and sensitivity analysis II: structure shape variation. Int J Solids Struct 20(6):527–552MathSciNetCrossRefGoogle Scholar
- Dems K, Mroz Z (1985) Variational approach to first-and second-order sensitivity analysis of elastic structures. Int J Numer Methods Eng 21(4):637–661MathSciNetCrossRefGoogle Scholar
- Dems K, Mroz Z (1987) Variational approach to sensitivity analysis in thermoelasticity. J Therm Stresses 10(4):283–306MathSciNetCrossRefGoogle Scholar
- Haftka R, Gürdal Z (2012) Elements of structural optimization, vol 11. Springer Science & Business Media, DordrechtzbMATHGoogle Scholar
- Haftka R, Mroz Z (1986) First-and second-order sensitivity analysis of linear and nonlinearstructures. AIAA J 24(7):1187–1192MathSciNetCrossRefGoogle Scholar
- Kaessmair S, Steinmann P (2018) Computational mechanics of generalized continua. Springer, Berlin/Heidelberg, pp 1–13. https://doi.org/10.1007/978-3-662-53605-6_111-1 Google Scholar
- Lekszycki T (2018) Variational methods in optimization of structures, Methods of. Springer, Berlin/Heidelberg, pp 1–9Google Scholar
- Mróz Z, Piekarski J (1998) Sensitivity analysis and optimal design of non-linear structures. Int J Numer Methods Eng 42(7):1231–1262MathSciNetCrossRefGoogle Scholar
- Pedersen P, Cheng G, Rasmussen J (1989) On accuracy problems for semi-analytical sensitivity analyses. J Struct Mech 17(3):373–384Google Scholar
- Smith D, Tortorelli D, Tucker III C (1998a) Analysis and sensitivity analysis for polymer injection and compression molding. Comput Methods Appl Mech Eng 167(3–4):325–344CrossRefGoogle Scholar
- Smith D, Tortorelli D, Tucker III C (1998b) Optimal design for polymer extrusion. Part I: sensitivity analysis for nonlinear steady-state systems. Comput Methods Appl Mech Eng 167(3–4):283–302CrossRefGoogle Scholar
- Smith D, Tortorelli D, Tucker III C (1998c) Optimal design for polymer extrusion. Part II: sensitivity analysis for weakly-coupled nonlinear steady-state systems. Comput Methods Appl Mech Eng 167(3–4):303–323CrossRefGoogle Scholar
- Tortorelli D, Michaleris P (1994) Design sensitivity analysis: overview and review. Inverse Prob Eng 1(1): 71–105CrossRefGoogle Scholar
- Tortorelli D, Subramani G, Lu S, Haber R (1991) Sensitivity analysis for coupled thermoelastic systems. Int J Solids Struct 27(12):1477–1497CrossRefGoogle Scholar
- Turco E, dell Isola F, Cazzani A, Rizzi N (2016) Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4):85Google Scholar
- Van Keulen F, Haftka R, Kim N (2005) Review of options for structural design sensitivity analysis. Part 1: linear systems. Comput Methods Appl Mech Eng 194:3213–3243CrossRefGoogle Scholar