Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Sensitivity Analysis in Structural and Multidisciplinary Problems

  • Tomasz LekszyckiEmail author
  • Fabio Di Cosmo
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_265-1
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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

As a starting point let \(\mathcal {B}_0\) be the reference configuration of a continuous body, described as a compact domain of a three-dimensional Euclidean affine space \(\mathcal {E}^3\), with boundary \(\partial \mathcal {B}_0 = \varOmega _0\). From a kinematical point of view, the placement of the body is represented by the field \(\chi \, : \, \mathcal {B}_0 \, \rightarrow \, \mathcal {B}\subset \mathcal {E}^3\), which, for the sake of simplicity, will be supposed twice differentiable. Its mechanical behavior is described as a Cauchy continuum governed, therefore, by the balance equations for mass and momentum (this is just a simplified assumption, since one could consider also higher gradient models, i.e., models in which the energy could depend also on higher gradient of the placement field and not only on its first gradient dell’Isola et al. (2018)). In particular, if one is interested in the equilibrium configurations of the system, they are solutions of the following equation:
$$\displaystyle \begin{aligned} \nabla \sigma + \mathbf{f} = 0 {} \end{aligned} $$
(1)
subject to the boundary conditions:
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \mathbf{u}={\mathbf{u}}_0 \quad \mathrm{on} \; \varOmega_1 {} \end{array} \end{aligned} $$
(2)
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \mathbf{t}= \sigma \cdot {\mathbf{n}}_0 = {\mathbf{t}}^{(e)} \quad \mathrm{on} \; \varOmega_2 {}, \end{array} \end{aligned} $$
(3)
where Ω1 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 Ω2 is the region of the boundary where the external traction is assigned, and Ω0 = Ω1 + Ω2. Here X denotes the coordinate of a material particle in the reference configuration \(\mathcal {B}_0\), and n0 is the normal vector to the surface Ω0. 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:
$$\displaystyle \begin{aligned} \left< \delta \mathsf{E}, \sigma\right>_{\mathcal{B}_0} = \left< \delta \mathbf{u}, \mathbf{f} \right>_{\mathcal{B}_0} + \left\langle {\mathbf{t}}^{(e)}, \delta\mathbf{u} \right\rangle_{\varOmega_0}\,. {} \end{aligned} $$
(4)
Here and in the rest of the entry, the brackets \(\left < \,\cdot ,\, \cdot \right >_{\cdot }\) denote the product between the argument fields integrated over the domain specified by the subscript.
At this point a useful digression can be inserted. Indeed, a wider class of phenomena can be described by means of a linear differential operator L, whose action on the displacement field u will be denoted by L(u). Therefore, one could replace Eq. 1 by the expression:
$$\displaystyle \begin{aligned} L(\mathbf{u})=-\mathbf{f}\,. {} \end{aligned} $$
(5)
Multiplying the above equation by δu, 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:
$$\displaystyle \begin{aligned} &\left\langle L(\mathbf{u}) , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0} {=} \left\langle -\mathbf{f} , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0} \\ &\left\langle \mathbf{u} , L^{a}(\delta\mathbf{u}) \right\rangle_{\mathcal{B}_0} {-} \left\langle B(\mathbf{u}) , B^a(\delta\mathbf{u}) \right\rangle_{\varOmega_0}\\ &\qquad {=} \left\langle -\mathbf{f} , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0}\,, \end{aligned} $$
where La represents the adjoint differential operator associated with L and the chosen scalar product. The linear operators B and Ba 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:
$$\displaystyle \begin{aligned} \left\langle B(\mathbf{u}), B^a(\delta\mathbf{u}) \right\rangle_{\varOmega_0} & = \left\langle B(\mathbf{u}) , B^a(\delta\mathbf{u}) \right\rangle_{\varOmega_1}\\ &\quad +\left\langle B(\mathbf{u}) , B^a(\delta\mathbf{u}) \right\rangle_{\varOmega_2}\,. \end{aligned} $$
(6)
The first term of the right-hand side vanishes because of the imposed boundary conditions on the displacement field, whereas the second term represents the work done by the external traction on the admissible variation δu along the surface Ω2.
In the case of a linear elastic body, it is possible to write:
$$\displaystyle \begin{aligned} &\left\langle \nabla \cdot \sigma , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0} = \left\langle -\mathbf{f} , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0} \\ &\left\langle \sigma , \frac{1}{2}\left(\nabla\delta\mathbf{u} + \left( \nabla \delta \mathbf{u} \right)^T \right) \right\rangle_{\mathcal{B}_0} - \left\langle \sigma \cdot {\mathbf{n}}_0 , \delta\mathbf{u} \right\rangle_{\varOmega_2}\\ &\qquad = \left\langle \mathbf{f} , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0}\\ &\left\langle \sigma , \delta \mathsf{E} \right\rangle_{\mathcal{B}_0} = \left\langle {\mathbf{t}}^{(e)} , \delta\mathbf{u} \right\rangle_{\varOmega_2} + \left\langle \mathbf{f} , \delta\mathbf{u} \right\rangle_{\mathcal{B}_0}\,, \end{aligned} $$
which coincides with Eq. (4).
In a big class of sensitivity analysis problems, one is interested in the dependence of a set of objective functionals, Kj, which describe some properties of the body under investigation, with respect to design variables, which in this section will be described as continuous fields bp(⋅) (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 Kj with respect to design parameters, i.e.:
$$\displaystyle \begin{aligned} \frac{\delta K_j}{\delta b_p} = \frac{d}{d\epsilon} Kj\left( b_p + \epsilon \delta b_p \right)\mid_{\epsilon = 0} \,. \end{aligned}$$
These quantities are usually called first-order sensitivities. For the sake of simplicity, the case of a single objective functional, K, will be analyzed in the rest of the entry. In particular this functional is written as the sum of four terms:
$$\displaystyle \begin{aligned} K & = \int_{\mathcal{B}_0}g_0(\sigma,b_p)dV + \int_{\mathcal{B}_0}g_1(\mathbf{u},b_p)dV\\ &\quad + \int_{\partial \mathcal{B}_0^E}g_2(\mathbf{t})d\varSigma + \int_{\partial \mathcal{B}_0^N}g_3(\mathbf{u})d\varSigma\,, {} \end{aligned} $$
(7)
where the dependence on the design variables bp(⋅) is both explicit, in g0 and g1, 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

The direct method for computing first-order design sensitivity is the most intuitive one since it involves the direct evaluation of the sensitivity of the stress and displacement fields. Indeed, assuming that the functions g0, g1, g2, g3 are all differentiable with respect to their arguments, the first derivative of the functional (7) can be expressed as follows:
$$\displaystyle \begin{aligned} \frac{\delta K}{\delta b_p} & = \int_{\mathcal{B}_0}\left( \frac{\partial g_0}{\partial \sigma}\frac{\delta \sigma}{\delta b_p } + \frac{\partial g_0}{\partial b_p} \right) dV+\\ &\quad + \int_{\mathcal{B}_0}\left( \frac{\partial g_1}{\partial \mathbf{u}}\frac{\delta \mathbf{u}}{\delta b_p } + \frac{\partial g_1}{\partial b_p} \right) dV + \end{aligned} $$
(8)
$$\displaystyle \begin{aligned} &\quad + \int_{\partial \mathcal{B}_0^E}\left( \frac{\partial g_2}{\partial \mathbf{t}} \frac{\delta \mathbf{t}}{\delta b_p} \right) d\varSigma+\\ &\quad + \int_{\partial \mathcal{B}_0^N}\left( \frac{\partial g_3}{\partial \mathbf{u}} \frac{\delta \mathbf{u}}{\delta b_p} \right) d\varSigma\,. {} \end{aligned} $$
(9)
It can be immediately noticed that, in order to obtain the final result, one needs to know the sensitivities \(\frac {\delta \mathbf {u}}{\delta b_p}\) and \(\frac {\delta \sigma }{\delta b_p}\). Differentiation of the equilibrium equation (1) with respect to the design variables leads to the following equation for the unknown sensitivity of the stress field:
$$\displaystyle \begin{aligned} \left< \delta \mathsf{E}, \frac{\delta \sigma}{\delta b_p}\right>_{\mathcal{B}_0} = 0\,. {} \end{aligned} $$
(10)
The additional information about the relationship between stress and displacement is given by the constitutive relations and their differentiation with respect to the design variables. In the case of linear elasticity, for instance, one has that:
$$\displaystyle \begin{aligned} \sigma = C : \mathsf{E}\,, {} \end{aligned} $$
(11)
where C is the elasticity tensor and the symbol:  denotes the double contraction between tensors. Therefore, the derivative of the stress tensor can be written as follows:
$$\displaystyle \begin{aligned} \frac{\delta \sigma}{\delta b_p}= C: \nabla\frac{\delta \mathbf{u}}{\delta b_p} + \frac{\delta C}{\delta b_p}: \mathsf{E}\,. {} \end{aligned} $$
(12)
Substituting Eq. 12 into Eq. 10, one can obtain an equation for the unknown sensitivity \(\frac {\delta \mathbf {u}}{\delta b_p}\) which is similar to Eq. 4. The boundary conditions, instead, become:
$$\displaystyle \begin{aligned} \frac{\delta \mathbf{t}}{\delta b_p} &= \frac{\delta \sigma}{\delta b_p} \cdot \mathbf{n} = 0 \quad \mathrm{on}\;\varOmega_2 \end{aligned} $$
(13)
$$\displaystyle \begin{aligned} \frac{\delta \mathbf{u}}{\delta b_p} &= 0 \quad \mathrm{on} \; \varOmega_1\,. \end{aligned} $$
(14)
However, due to the differentiation of the constitutive relationship, some additional terms, which can be interpreted as mechanical loads, have been produced. Eventually it is possible to write the following equations:
$$\displaystyle \begin{aligned} & \left\langle \delta \mathbf{u}, \nabla \left( C:\nabla \frac{\delta \mathbf{u}}{\delta b_p} \right) \right\rangle_{\mathcal{B}_0} - \left\langle \delta \mathbf{u} , \left( C:\nabla \frac{\delta \mathbf{u}}{\delta b_p} \right)\cdot {\mathbf{n}}_0 \right\rangle_{\varOmega_2} = \left\langle \delta\mathbf{u} , -\nabla \cdot \left( \frac{\delta C}{\delta b_p}: \mathsf{E} \right) \right\rangle_{\mathcal{B}_0} + \\ & + \left\langle \delta \mathbf{u}, \left( \frac{\delta C}{\delta b_p}: \mathsf{E} \right)\cdot {\mathbf{n}}_0 \right\rangle_{\varOmega_2}\,, \end{aligned} $$
(15)
where the right-hand side contains terms which can be interpreted as additional loads.

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).

Let H be the following extended functional:
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle H = K + \left< \mathsf{E}^a, \left( \sigma - C:\mathsf{E} \right) \right>_{\mathcal{B}_0} + \left< \sigma^a, \left( \mathsf{E} - \nabla \mathbf{u} \right) \right>_{\mathcal{B}_0} - \frac{1}{2} \left< \sigma, \left( \nabla{\mathbf{u}}^a + \left( \nabla {\mathbf{u}}^a\right)^T \right) \right>_{\mathcal{B}_0} +\\ &\displaystyle + \left< \mathbf{f}, {\mathbf{u}}^a \right>_{\mathcal{B}_0} + \int_{\partial \mathcal{B}_0}\mathbf{t}\cdot {\mathbf{u}}^a d\varSigma\,, {} \end{array} \end{aligned} $$
where the fields with the additional index (⋅)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:
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \frac{\delta H}{\delta b_p} = \int_{\mathcal{B}_0}\left[ \left( \frac{\partial g_0}{\partial \sigma} + \mathsf{E}^a - \frac{1}{2}\left( \nabla{\mathbf{u}}^a + \left( \nabla {\mathbf{u}}^a\right)^T \right) \right)\frac{\delta \sigma}{\delta b_p} + \left( \sigma^a-C:\mathsf{E}^a \right) + \right. \\ &\displaystyle + \left. \left( \nabla \sigma^a + \frac{\partial g_1}{\partial \mathbf{u}}\right) \frac{\delta u}{\delta b_p} + \mathsf{E}^a:\frac{\partial C}{\partial b_p}: \mathsf{E} + \left( \frac{\partial g_0}{\partial b_p} + \frac{\partial g_1}{\partial b_p} \right) \right]dV + \\ &\displaystyle \int_{\partial \mathcal{B}_0^E} \frac{\delta \mathbf{t}}{\delta b_p}\left( {\mathbf{u}}^a - \frac{\partial g_2}{\partial \mathbf{t}} \right)d\varSigma + \int_{\partial \mathcal{B}_0^N} \frac{\delta \mathbf{u}}{\delta b_p} \left( \frac{\partial g_3}{\partial \mathbf{u}}- n\cdot \sigma^a \right)d\varSigma + \\ &\displaystyle + \int_{\mathcal{B}_0}\left[ \frac{\delta \sigma^a}{\delta b_p} \left( E - \frac{1}{2} \left( \nabla\mathbf{u} + \left( \nabla \mathbf{u}\right)^T \right) \right) + \frac{\delta \mathsf{E}^a}{\delta b_p} \left( \sigma - C:\mathsf{E} \right) + \frac{\delta {\mathbf{u}}^a}{\delta b_p} \left( \nabla \sigma + \mathbf{f} \right) \right] dV\,. \end{array} \end{aligned} $$
(16)
If the primary fields satisfy the equations for the equilibrium of the elastic body, the last line in the above variation can be eliminated. On the other hand, it is possible to choose the adjoint fields such that they satisfy the following system of equations:
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \mathsf{E}^a = \frac{\partial g_0}{\partial \sigma} - \frac{1}{2}\left( \nabla{\mathbf{u}}^a + \left( \nabla {\mathbf{u}}^a\right)^T \right)\\ &\displaystyle \sigma^a = C:\mathsf{E}^a\\ &\displaystyle \quad \nabla \sigma^a + \frac{\partial g_1}{\partial \mathbf{u}} =0 \\ &\displaystyle n\cdot \sigma^a = \frac{\partial g_3}{\partial \mathbf{u}} \quad \mathrm{on}\;\partial \mathcal{B}_0^N \\ &\displaystyle {\mathbf{u}}^a = \frac{\partial g_2}{\partial \mathbf{t}} \quad \mathrm{on}\;\partial \mathcal{B}_0^E\,. \end{array} \end{aligned} $$
It can be immediately noticed that \(\frac {\partial g_0}{\partial \sigma }\) plays the role of an initial strain, whereas \(\frac {\partial g_3}{\partial \mathbf {u}}\) and \(\frac {\partial g_2}{\partial \mathbf {t}}\) provide the two boundary conditions for the adjoint structure.
Inserting the two solutions of the primary and adjoint problems in the expression of the sensitivity, one obtains the final result: where all the quantities inside the square brackets are known. It is clear from the above discussion that for every additional objective functional one obtains an additional adjoint system. Therefore the number of additional solutions to find increases with the number of objective functionals, whereas the cost of the direct method increases with the number of design variables. Therefore, the choice between the two methods depends on the ratio of the number of objective functionals to the number of design variables.

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).

After applying discretization techniques, the equilibrium equation can be rewritten in terms of the nodal displacement vector u as follows:
$$\displaystyle \begin{aligned} \mathsf{K}\mathbf{u} = \mathbf{f}\,, {} \end{aligned} $$
(18)
where K is the stiffness matrix and 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:
$$\displaystyle \begin{aligned} \frac{d G}{d b_p} = \frac{\partial G}{\partial b_p} + \frac{\partial G}{\partial \mathbf{u}} \frac{d \mathbf{u}}{d b_p}\,. {} \end{aligned} $$
(19)
In order to evaluate the sensitivity of the objective function, the sensitivity\(\frac {d\mathbf {u}}{db_p}\) of the nodal vector is required. Also for the discrete analysis, this quantity can be computed using two different approaches, the direct method and the adjoint one.

Direct Method for Discrete Sensitivity Analysis

Differentiating the equilibrium condition Eq. 18 with respect to the design variables, one obtains the following equation:
$$\displaystyle \begin{aligned} \mathsf{K}\frac{d\mathbf{u}}{db_p} = -\frac{d\mathsf{K}}{db_p}\mathbf{u} + \frac{d\mathbf{f}}{db_p}\,. {} \end{aligned} $$
(20)
If K is invertible (this is true when K is positive definite), the solution to this problem can be easily written as follows:
$$\displaystyle \begin{aligned} \frac{d\mathbf{u}}{db_p} = \mathsf{K}^{-1}\left( \frac{d\mathbf{f}}{db_p} - \frac{d\mathsf{K}}{db_p}\mathbf{u} \right)\,. \end{aligned} $$
(21)
Finally, by replacing this expression in Eq. 19, one obtains the following expression for the sensitivity analysis of the function G:
$$\displaystyle \begin{aligned} \frac{d G}{d b_p} = \frac{\partial G}{\partial b_p} + \frac{\partial G}{\partial \mathbf{u}}\mathsf{K}^{-1}\left( \frac{d\mathbf{f}}{db_p} - \frac{d\mathsf{K}}{db_p}\mathbf{u} \right)\,. \end{aligned} $$
(22)
This very simple method only requires the invertibility of the stiffness matrix (a condition which is ensured in finite element analysis) and the computation of the derivatives of K and 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 \(\frac {dG}{db_p}\).

Let H be the extended function:
$$\displaystyle \begin{aligned} H = G - \lambda^T \left( \mathsf{K}\mathbf{u} - \mathbf{f} \right)\,, \end{aligned} $$
(23)
where 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:
$$\displaystyle \begin{aligned} \frac{dH}{db_p}&= \frac{\partial G}{\partial b_p} + \frac{\partial G}{\partial \mathbf{u}} \frac{d \mathbf{u}}{d b_p} - \frac{d\lambda^T}{db_p}\left( \mathsf{K}\mathbf{u} - \mathbf{f} \right)\\ &\quad - \lambda^T\left( \mathsf{K}\frac{d\mathbf{u}}{db_p} + \frac{d\mathsf{K}}{db_p}\mathbf{u} - \frac{d\mathbf{f}}{db_p} \right) \end{aligned} $$
(24)
The nodal vector 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:
$$\displaystyle \begin{aligned} \mathsf{K}\lambda = \frac{\partial G}{\partial \mathbf{u}}\,. {} \end{aligned} $$
(25)
According to this choice, the following result becomes a straightforward consequence:
$$\displaystyle \begin{aligned} \frac{dH}{db_p}=\frac{dG}{db_p}= \frac{\partial G}{\partial b_p} - \lambda^T\left( \frac{d\mathsf{K}}{db_p}\mathbf{u} - \frac{d\mathbf{f}}{db_p} \right)\,. \end{aligned} $$
(26)
It is worth remarking once more that this sensitivity is expressed only in terms of the vectors 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

The last part of this section will be devoted to a short digression about sensitivity analysis for nonlinear equilibrium equations (more details can be found in Choi and Kim 2006b; Haftka and Mroz 1986; Mróz and Piekarski 1998; Cardoso and Arora 1988, for instance). In nonlinear mechanical problems, the equilibrium condition can be expressed as follows:
$$\displaystyle \begin{aligned} q(\mathbf{u},\mathbf{b}) = \mu \mathbf{f}(\mathbf{b})\,, {} \end{aligned} $$
(27)
where 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:
$$\displaystyle \begin{aligned} \mathsf{J}\frac{d\mathbf{u}}{db_p} = \mu \frac{d\mathbf{f}}{db_p} - \frac{\partial q}{\partial b_p}\,, \end{aligned} $$
(28)
where the tangent stiffness matrix is
$$\displaystyle \begin{aligned} \mathsf{J}=\frac{\partial q}{\partial \mathbf{u}}\,. \end{aligned}$$
Efficient approximation for the solution u and the tangent stiffness J can be computed by means of Newton’s iterative methods (see Haftka and Gürdal 2012). Therefore one gets:
$$\displaystyle \begin{aligned} \frac{d\mathbf{u}}{db_p}=\mathsf{J}^{-1}\left( \mu \frac{d\mathbf{f}}{db_p} - \frac{\partial q}{\partial b_p} \right)\,, \end{aligned} $$
(29)
and this expression can be replaced in Eq. 19 to obtain the final expression:
$$\displaystyle \begin{aligned} \frac{dG}{db_p} = \frac{\partial G}{\partial b_p} + \frac{\partial G}{\partial \mathbf{u}}\mathsf{J}^{-1}\left( \mu \frac{d\mathbf{f}}{db_p} - \frac{\partial q}{\partial b_p} \right)\,. \end{aligned} $$
(30)
Concerning the adjoint method, it proceeds as in the previous section, but replacing the stiffness K with the tangent stiffness J. In particular one obtains the following expression:
$$\displaystyle \begin{aligned} \frac{dG}{db_p} = \frac{\partial G}{\partial b_p} + \lambda^T\left( \mu \frac{d\mathbf{f}}{db_p} - \frac{\partial q}{\partial b_p} \right)\,, \end{aligned} $$
(31)
where the vector λ is solution of the adjoint linear problem:
$$\displaystyle \begin{aligned} \mathsf{J}^T\lambda = \frac{\partial G}{\partial \mathbf{u}}\,. \end{aligned}$$

Example: A Vibrating Beam with an Elastic Foundation

In order to show how the procedure can be implemented, an easy example will be investigated in this section. The system under investigation is a linear Euler-Bernoulli beam, clamped at the left end, over a spring foundation under a cyclic load. The equation of motion for the system is:
$$\displaystyle \begin{aligned} \left[ EJy'' \right]'' + K(x)y + A\rho \ddot{y} = F_0e^{i\omega t}\,, \end{aligned} $$
(32)
where 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 ac = 2 cm and area A = 4 cm2, 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 F0 = 10N, and its frequency is \(\nu = \frac {\omega }{2\pi }=40\) Hz.
The time dependence in the problem can be factorized choosing a solution in the form:
$$\displaystyle \begin{aligned} y(x,t)=u(x)\mathrm{e}^{i\omega t}\,, {} \end{aligned} $$
(33)
which gives the following equation for the amplitude u(x):
$$\displaystyle \begin{aligned} \frac{d^2}{dx^2}\left[ EJ\frac{d^2u}{dx^2} \right] + K(x)u - A\rho \omega^2 u = F_0\,. {} \end{aligned} $$
(34)
Let the objective functional be \(G=u(s)=\int _{0}^L \delta (x-s)u(x)dx\) the displacement of the material particle occupying the position 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 = s0.
The sensitivity of the functional 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:
$$\displaystyle \begin{aligned} H = \int_0^L \delta (x-s)u(x)dx - \int_0^L v^a(x)\left( \frac{d^2}{dx^2}\left[ EJ\frac{d^2u}{dx^2} \right] + K(x)u - A\rho \omega^2 u - F_0 \right)\,, \end{aligned} $$
where the boundary term is not included since the adjoint field va(⋅) 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:
$$\displaystyle \begin{aligned} \frac{\delta H}{\delta K(s_0)} &= \int_0^L \left( \frac{d^2}{dx^2}\left[ EJ\frac{d^2v^{a}}{dx^2} \right]+\right.\\ &\quad + K(x)v^a - A\rho \omega^2 v^a-\\ & \quad - \delta(x-s) \bigg)\frac{\delta u}{\delta K(x_0)}dx + \\ &\quad + u(s_0)v^a(s_0)\,.{} \end{aligned} $$
(35)
By choosing as adjoint field va(⋅) the solution of the equation:
$$\displaystyle \begin{aligned} \frac{d^2}{dx^2}\left[ EJ\frac{d^2v^{a}}{dx^2} \right] + K(x)v^a - A\rho \omega^2 v^a = \delta(x-s)\,, \end{aligned} $$
(36)
the sensitivity of the objective functional has the following expression:
$$\displaystyle \begin{aligned} \frac{\delta G}{\delta K(s_0)} = u(s_0)v^a(s_0)\,, \end{aligned} $$
(37)
which is written only in terms of the primary and adjoint displacement fields. Let us notice that, in this situation, the adjoint system is again a clamped beam with the same features of the primary one but the load is different: It is a unit dead load at the material point, which in the reference configuration occupies the position x = s, which is also the material point involved in the definition of the objective functional G = u(s).
The numerical implementation of the problem has been performed using the weak form package of the software ComsolMultiphysics, which uses standard finite element methods for solving the problem. Both the primary and the adjoint solutions, u(⋅) and va(⋅), 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 s0 have been varied.
Fig. 1

Plot of the sensitivity \( \frac {\delta G}{\delta K(s_0)}\) for some values of the parameter s and with s0ranging from s0 = 0.1 to s0 = 1.99 m

Fig. 2

Plot of the maximal and minimal values of the sensitivity \( \frac {\delta G}{\delta K(s_0)}\) as the parameters s varies from s = 0.1 to s = 1.99 m

In particular the chart plotted in Fig. 1 shows the value of the sensitivity as a function of the position s0 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 s0 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.

Let the discrete system be described by the nodal displacement vector u obeying the equilibrium condition:
$$\displaystyle \begin{aligned} \mathsf{K}\mathbf{u}=\mathbf{f}\,. \end{aligned} $$
(38)
The objective function is G(u, b) and its second derivative can be easily computed:
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \frac{d^2G}{db_pdb_r}= \frac{\partial^2 G}{\partial b_p \partial b_r} + \frac{\partial^2 G}{\partial u_j \partial b_r}\frac{du_j}{db_p} + \frac{\partial^2 G}{\partial u_j \partial b_p}\frac{du_j}{db_r} + \\ &\displaystyle +\frac{\partial^2 G}{\partial u_j \partial u_l}\frac{du_j}{db_p}\frac{du_l}{db_r}+ \frac{\partial G}{\partial u_j }\frac{d^2 u_j}{db_rdb_p}\,. {} \end{array} \end{aligned} $$
(39)
This expression involves the unknown second-order sensitivities \(\frac {d^2 u_j}{db_rdb_p}\) which can be computed by deriving twice the equilibrium condition:
$$\displaystyle \begin{aligned} \mathsf{K}\frac{d\mathbf{u}}{db_pdb_r} &= \frac{\partial^2 \mathbf{f}}{\partial b_p \partial b_r} + \frac{\partial \mathsf{K}}{\partial b_p}\frac{d\mathbf{u}}{db_r} + \frac{\partial \mathsf{K}}{\partial b_r}\frac{d\mathbf{u}}{db_p}+\\ &\quad + \frac{\partial^2 \mathsf{K}}{\partial b_p \partial b_r}\mathbf{u}\,. \end{aligned} $$
(40)
If the matrix K is invertible, it is straightforward to find the expression for the unknown \(\frac {d^2\mathbf {u}}{db_pdb_r}\), and consequently a simple replacement of the solution in Eq. 39 provides the desired expression for the second-order sensitivity of the function G. Despite its simplicity, this method is numerically costly since it grows quadratically in the number of design variables.
A more efficient approach in this case is the hybrid direct-adjoint method. Starting from the expression of the first-order sensitivity analysis:
$$\displaystyle \begin{aligned} \frac{dG}{db_p}= \frac{\partial G}{\partial b_p} - \lambda^T\left( \frac{d\mathsf{K}}{db_p}\mathbf{u} - \frac{d\mathbf{f}}{db_p} \right)\,, \end{aligned} $$
(41)
and differentiating again with respect to the design variables, one obtains the following expression for the second-order sensitivity analysis:
$$\displaystyle \begin{aligned} \frac{d^2 G}{db_pdb_r} & = \frac{\partial^2G}{\partial b_p \partial b_r} + \frac{\partial G}{\partial b_p \partial u_j}\frac{du_j}{db_r}\\ &\quad - \frac{d\lambda^T}{db_r}\left( \frac{\partial\mathsf{K}}{\partial b_p}\mathbf{u} - \frac{d\mathbf{f}}{db_p} \right) +\\ &\quad - \lambda^T \left( \frac{\partial \mathsf{K}}{\partial b_p}\frac{d\mathbf{u}}{db_r} + \frac{\partial^2 \mathsf{K}}{\partial b_p \partial b_r}\mathbf{u}\right.\\ &\quad \left. -\frac{\partial^2 \mathbf{f}}{\partial b_p \partial b_r} \right)\,. \end{aligned} $$
By deriving Eq. 25 with respect to the design variables, one obtains the following equation:
$$\displaystyle \begin{aligned} \mathsf{K}\frac{d\lambda_j}{db_r} = \frac{\partial^2 G}{\partial u_j \partial u_k }\frac{du_k}{db_r} - \frac{\partial\mathsf{K}_{jl}}{\partial b_r}\lambda_l\,, \end{aligned} $$
(42)
which can be replaced in the expression of the second-order sensitivity analysis to write the final result:
$$\displaystyle \begin{aligned} \frac{d^2 G}{db_pdb_r}& = \frac{\partial^2G}{\partial b_p \partial b_r} + \frac{\partial G}{\partial b_p \partial u_j}\frac{du_j}{db_r}-\\ &\quad - \frac{du_j}{db_r}\frac{\partial^2 G}{\partial u_j\partial u_k}\frac{du_k}{db_p} + \\ &\quad - \lambda^T \left( \frac{\partial \mathsf{K}}{\partial b_p}\frac{d\mathbf{u}}{db_r} + \frac{\partial \mathsf{K}}{\partial b_r}\frac{d\mathbf{u}}{db_p}\right.\\ &\quad \left.+ \frac{\partial^2 \mathsf{K}}{\partial b_p \partial b_r}\mathbf{u} -\frac{\partial^2 \mathbf{f}}{\partial b_p \partial b_r} \right)\,. \end{aligned} $$
If the sensitivities \(\frac {d\mathbf {u}}{db_p}\) are computed by the direct method, all the quantities that appear in the right-hand side of the above expression are known, and this result requires to find 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

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Production EngineeringWarsaw University of TechnologyWarszawaPoland
  2. 2.International Research Center M& University of L’AquilaL’AquilaItaly

Section editors and affiliations

  • Francesco Dell’Isola
    • 1
    • 2
  1. 1.International Research Center M&MoCS, University of L’AquilaL’AquilaItaly
  2. 2.DISGUniversità di Roma “La Sapienza”RomaItaly