Shape gradients for the failure probability of a mechanic component under cyclic loading: a discrete adjoint approach

Abstract

This work provides a numerical calculation of shape gradients of failure probabilities for mechanical components using a first discretize, then adjoin approach. While deterministic life prediction models for failure mechanisms are not (shape) differentiable, this changes in the case of probabilistic life prediction. The probabilistic, or reliability based, approach thus opens the way for efficient adjoin methods in the design for mechanical integrity. In this work we propose, implement and verify a method for the numerical calculation of the shape gradients of failure probabilities for the failure mechanism low cycle fatigue, which applies to polycrystalline metal. Numerical examples range from a bended rod to a complex geometry from a turbo charger in 3D.

Detailed calculations

1.1 Notation for finite elements

To fix the notation for later purposes, we introduce the standard discretization of the PDE (1) with Lagrange finite elements [14, 16].

Let the finite element be given by $\{K,P(K),{\varSigma }(K)\}$ where $K\subseteq {\mathbb {R}}^3$ is a compact, connected Lipschitz set, P is a finite vector space polynomials $p:K\rightarrow {\mathbb {R}}$. The local degrees of freedom are linear functionals on P(K) defined by $\varphi _{K,j}(p)=p(X_j)$ where there are nodes $X_{1}^K,\ldots ,X_{n_{\mathrm{sh}}}^K\in K$ such that the mapping from P(K) to its local degrees of freedom is bijective. The dual basis of P(K) with respect to the local degrees of freedom are the local shape functions $\theta _{K,k}\in P(K)$ and $\varphi _{K,j}(\theta _{K,k})=\delta _{j,k}$, $j,k\in \{1,\ldots ,n_\mathrm{sh}\}$.

Let ${\mathcal {T}}_h$ be a finite element mesh on ${\varOmega }$ with $N_{\mathrm{el}}$ elements. As usually we assume that there is a reference element $\{{\widehat{K}},{\widehat{P}}, {\widehat{{\varSigma }}})$ with a finite dimensional linear space of reference polynomials ${\widehat{P}}$. For each element $K\in {\mathcal {T}}_h$ in the mesh, we assume that there is a bijective transformation $T_K:{{\widehat{K}}}\rightarrow K$ such that ${\widehat{P}}=P\circ T_K$ , ${{\widehat{\theta }}}_{j}=\theta _{j}\circ T_K$ and ${{\widehat{\varphi }}}_{j}(p\circ T_K)=\varphi _{j}(p)$, $j\in \{1,\ldots ,n_{\mathrm{sh}}\}$. In many cases, we have

$$\begin{aligned} T_K({\hat{\xi }})=T_K({\hat{\xi }},X)=\sum _{j=1}^{n_\mathrm{sh}}{{\widehat{\theta }}}_j({\widehat{\xi }})X_{j}^K,~~{{\widehat{\xi }}}\in {\widehat{K}}. \end{aligned}$$

(26)

Let $X=\{X_1,...,X_N\}=\bigcup _{K\in {\mathcal {T}}_h}\{X_{1}^K,...,X_{n_{\mathrm{sh}}}^K\}$ be the set of all the Lagrange nodes. For $K\in {\mathcal {T}}_h$ and $m\in \{1,...,n_\mathrm{sh}\}$, let ${\widehat{j}}(K,m)\in \{1,...,N\}$ be the corresponding index of the Lagrange node. The mapping ${\widehat{j}}(\cdot ,\cdot ):{\mathcal {T}}_h\times \{1,\ldots ,n_\mathrm{sh}\}\rightarrow \{1,\ldots ,N\}$ is also called the connectivity.

Let $\{\theta _{j}:j\in \{1,\ldots ,N\}\}$, be the set of global shape functions $\theta _{j}:{\varOmega }\rightarrow {\mathbb {R}}$. When these functions are restricted to $K\in {\mathcal {T}}_h$, they belong to P(K) and fulfill

(27)

The global finite element space $H_h^1({\varOmega },{\mathbb {R}})$ is the linear span of $\{\theta _{j}:j\in \{1,\ldots ,N\}\}$. Let $H^1_h({\varOmega },{\mathbb {R}}^3)=H_h^1({\varOmega },{\mathbb {R}})^ {\times 3}$. We have $H_h^1({\varOmega },{\mathbb {R}}^3)\subseteq H^1({\varOmega },{\mathbb {R}}^3)$ and let $H^1_{D,h}({\varOmega },{\mathbb {R}}^3)$ be the subspace of $H^1_h({\varOmega },{\mathbb {R}}^3)$ with functions $u\in H_h^1({\varOmega },{\mathbb {R}}^3)$ vanishing on all boundary nodes $\overline{\partial {\varOmega }_D}\cap \{X_1,\ldots ,X_n\}$. If vanishing of u on the nodes in $\partial {\varOmega }_D$ implies vanishing of u on $\partial {\varOmega }_D$, we have $H_{D,h}^1({\varOmega },{\mathbb {R}}^3)=H_h^1({\varOmega },{\mathbb {R}}^3)\cap H^1_D({\varOmega },{\mathbb {R}}^3)$. A solution $u\in H_{D,h}^1({\varOmega },{\mathbb {R}}^3)$ to the discretized elasticity PDE then fulfills

$$\begin{aligned}&B(u,v)=\int \limits _{{\varOmega }}f\cdot v\, dx+\int \limits _{\partial {\varOmega }_N}g\cdot v \,dA , \nonumber \\&\quad \forall v\in H_{D,h}^1({\varOmega },{\mathbb {R}}^3). \end{aligned}$$

(28)

Furthermore, such a solution always exists by the coercivity of the bilinear form B(u, v), which also holds on $H^1_{D,h}({\varOmega },{\mathbb {R}}^3)$.

In the derivation of the adjoint equation in Sect. 4 we need explicit expressions for both sides of (28) in terms of the global degrees of freedom $U=(u_{j})_{j\in \{1,\ldots ,N\}}$, $u_j\in {\mathbb {R}}^3$, and the node coordinates X, where it is understood that $u_{j}=0$ if $X_j\in \partial {\varOmega }_D$. We rewrite (28) in terms of global variables U via

$$\begin{aligned} B(X)U&=F(X),~~B(X)_{(j,r),(k,s)}=B(e_r\theta _j,e_s\theta _k) \text{ and } \nonumber \\ F_{(j,r)}&=\int \limits _{{\varOmega }}f\cdot e_r\theta _j\, dx+\int \limits _{\partial {\varOmega }_N}g\cdot e_r\theta _j \,dA, \end{aligned}$$

(29)

with $e_r$, $r=1,2,3$ the standard Basis on ${\mathbb {R}}^ 3$. The linear equation (29) is understood in the sense that the (k, s) indices of the stiffness matrix B(X) are contracted with the related indices of U. The solution of this equation is denoted by U(X).

We have emphasised the dependency of the stiffness matrix B(X) and the load vector F(X) from the global node coordinate $N\times 3$-matrix X. The explicit calculations, which are completely standard, are given in “Appendix A.2” in order to prepare the ground for the calculation of shape sensitivities.

In some situations, also the surface and volume force densities $f=f(X)$ and $g=g(X)$ are allowed to have a differentiable dependency on X. For f this makes sense e.g. in the case of centrifugal loads and we might want to adapt a surface force density g under geometry changes, in order to keep the total force acting on a part of the surface fixed, even if the surface volume changes. For notational simplicity, we will only introduce this when needed for a physically correct description.

1.2 Discretization of the PDE

The integrals on both sides of (28) are implemented using a numerical quadrature with quadrature points ${\widehat{\xi }}_l$ and weights ${\widehat{\omega }}_l$, $l=1,\ldots ,l_q$, on the reference element ${\widehat{K}}$. We start with the discretization of the left hand side of (28):

$$\begin{aligned}&B(u,v)\nonumber \\&\quad =\lambda \sum _{K\in {\mathcal {T}}_h}\int \limits _{K} \nabla \cdot u\nabla \cdot v dx+2\mu \sum _{K\in {\mathcal {T}}_h}\int \limits _{K}\varepsilon (u):\varepsilon (v) dx \nonumber \\&\quad =\lambda \sum _{K\in {\mathcal {T}}_h}\int \limits _{{\widehat{K}}} \nabla \cdot u(T_K({\widehat{\xi }}))\nabla \cdot v(T_K({\widehat{\xi }}))\det ({\widehat{\nabla }}T_K({\widehat{\xi }})) d{\widehat{\xi }} \nonumber \\&\qquad +2\mu \sum _{K\in {\mathcal {T}}_h}\int \limits _{{\widehat{K}}} \varepsilon \bigl (u(T_K({\widehat{\xi }}))\bigr ):\varepsilon \bigl (v(T_K({\widehat{\xi }}))\bigr )\det ({\widehat{\nabla }}T_K({\widehat{\xi }})) d{\widehat{\xi }} \nonumber \\&\quad =\lambda \sum _{K\in {\mathcal {T}}_h}\sum _{l=1}^{l_q} \widehat{\omega _l} \det (\widehat{\nabla }T_K(\widehat{\xi _l}))\nabla \cdot u(T_K(\widehat{\xi _l}))\nabla \cdot v(T_K(\widehat{\xi _l})) \nonumber \\&\qquad +2\mu \sum _{K\in {\mathcal {T}}_h}\sum _{l=1}^{l_q}\widehat{\omega _l} \det ({\widehat{\nabla }}T_K(\widehat{\xi _l}))\varepsilon \bigl (u(T_K(\widehat{\xi _l}))\bigr ):\varepsilon \bigl (v(T_K(\widehat{\xi _l}))\bigr ) \end{aligned}$$

(30)

After setting $\omega _{lK}=\widehat{\omega _l} \det \left( {\widehat{\nabla }} T_K(\widehat{\xi _l})\right)$ and $\xi _l^K=T_K(\widehat{\xi _l})$, this can be written as

$$\begin{aligned} B(u,v)&=\underbrace{\lambda \sum _{K\in {\mathcal {T}}_h}\sum _{l=1}^{l_q}\omega _{lK}\nabla \cdot u(\xi _l^K)\nabla \cdot v(\xi _l^K)}_{B_1(u,v)} \nonumber \\&\quad +\underbrace{2\mu \sum _{K\in {\mathcal {T}}_h}\sum _{l=1}^{l_q}\omega _{lK}\varepsilon \bigl (u(\xi _l^K)\bigr ):\varepsilon \bigl (v(\xi _l^K)\bigr )}_{B_2(u,v)} \end{aligned}$$

(31)

If we apply (16) and (17) to u and v and insert the result into (31), we obtain the discretization of the left hand side.

For the volume integral on the right hand side of (28) we obtain by a similar argument

$$\begin{aligned} \int \limits _{{\varOmega }} f\cdot v\, dx=\sum _{K\in {\mathcal {T}}_h}\sum _{l=1}^{l_q} \omega _{lK} f(\xi _l^K)\cdot v(\xi _l^K). \end{aligned}$$

(32)

And finally we get the following expression for the discretized surface integral

$$\begin{aligned}&\int \limits _{\partial {\varOmega }}g\cdot v\,dA=\sum _{F\in {\mathcal {N}}_h}\sum _{l=1}^{l_q^F}\omega _{lF}g(\xi _{lF})\cdot v(\xi _{lF}),\text {where} \nonumber \\&\omega _{lF} ={\widehat{\omega }}^F_l\sqrt{\det g_F({\widehat{\xi }}_l^F)}~\text {and}~\xi _{l}^F=T_{K(F)} ({\widehat{\xi }}_l^F). \end{aligned}$$

(33)

Here ${\widehat{\xi }}_l^F$ and ${\widehat{\omega }}_l^F$ are quadrature points on a reference face ${{\widehat{F}}}$ of the reference element ${\widehat{K}}$ and ${\mathcal {N}}_h$ is the collection of all finite element faces that lie in $\partial {\varOmega }$. is the Gram matrix on ${\widehat{F}}$, where $T_K:{\widehat{F}}\rightarrow F$.

If the selected quadratures are not exact, the above identities have to be understood in the sense of approximations.

1.3 Computing $\frac{\partial J}{\partial U}$

In the following, we use some conventions related to the connectivity mapping ${\widehat{j}}:{\mathcal {T}}_h\times \{1,\ldots ,n_\mathrm{sh})\rightarrow \{1,\ldots ,N\}$. For $K\in {\mathcal {T}}_h$, we denote by ${\widehat{j}}_K:\{1,\ldots ,n_{\mathrm{sh}}\}\rightarrow \{1,\ldots ,N\}$ the restriction of the connectivity mapping ${\widehat{j}}$ to the set $\{(K,1),\ldots ,(K,n_k)\}$, where we identify $1,\ldots ,n_{\mathrm{sh}}$ with $(K,1),\ldots ,(K,{n_{\mathrm{sh}}})\}$.

For a fixed global index $j\in \{1,2,...,N\}$ we have ${\widehat{j}}^{-1}(j)=\{(K_1,m_1),...,(K_f,m_f)\}$, where $K_1,...,K_f \in {\mathcal {T}}_h$ and $m_1,...,m_f\in \{1,2,...,n_{\mathrm{sh}}\}$. With ${\widehat{j}}^{-1}(j)_1=\{K_1,\ldots ,K_f\}$ we denote the set projection to the first component.

For $k=1,2,3$ let $u_{jk}$ be the x, y, z coordinate of the global degree of freedom $u_j$. We obtain for the partial derivative of J with respect to this variable

$$\begin{aligned} \begin{aligned} \frac{\partial J(X,U)}{\partial u_{jk}}&= \sum _{\begin{array}{c} F\in {\mathcal {N}}_h \\ j\in {\widehat{j}}_{K(F)}(\{1,\ldots ,n_{\mathrm{sh}}\}) \end{array}}\sum _{l=1}^{l_q^F}\omega _{lF}\\&\quad \times \frac{\partial }{\partial u_{jk}}\left[ \mathrm{CMB}^{-1}\left( \varepsilon _{a}^{\mathrm{el-pl}}\biggl (\sum _{m=1}^{n_\mathrm{sh}}u_{{\widehat{j}}(K(F),m)}\otimes \right. \right. \\&\qquad \qquad \left. \left. (({\widehat{\nabla }} T_{K(F)}^{T}({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_m({\widehat{\xi }}_l^F))\biggr ) \right) \right] ^{-{\bar{m}}}. \end{aligned} \end{aligned}$$

We will frequently use the abbreviation

$$\begin{aligned} q&=q_{lF}=q_{lF}(U,X)=\nabla u(\xi _l^F) \nonumber \\&=\sum _{m=1}^{n_{\mathrm{sh}}}u_{j_{(K(F),m)}}\otimes ({\widehat{\nabla }} T_{K(F)}^{T}({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_m(\widehat{\xi _l})), ~l=1,\ldots ,l_q^ F, \end{aligned}$$

(34)

with $\xi _l^F=T_{K_F}({\hat{\xi }}_l)$. We thus have

$$\begin{aligned}&\frac{\partial }{\partial u_{jk}}\left[ \mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF}))\right] ^{-{\bar{m}}} \nonumber \\&\quad = -{\bar{m}}\left[ \mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF}))\right] ^{-{\bar{m}}-1}\nonumber \\&\qquad \,\, \times \frac{\partial }{\partial u_{jk}} \left[ \mathrm{CMB}^{-1}(\varepsilon _{a}^{\mathrm{el-pl}}(q_{lF}))\right] . \end{aligned}$$

(35)

In the next step, we compute

$$\begin{aligned} \frac{\partial }{\partial u_{jk}}[\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF}))]&= \frac{\frac{\partial }{\partial u_{jk}}\bigl (\varepsilon _{a}^\mathrm{el-pl}(q_{lF})\bigr )}{\frac{\partial \mathrm{CMB}}{\partial \varepsilon _a^{\mathrm{el-pl}}}(\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF})))} \end{aligned}$$

(36)

We now calculate

$$\begin{aligned} \frac{\partial }{\partial u_{jk}}\bigl (\varepsilon _{a}^\mathrm{el-pl}(q_{lF})\bigr )&= \frac{\partial \varepsilon _{a}^\mathrm{el-pl}(q_{lF})}{\partial q_{lF}}:\frac{\partial q_{lF}}{\partial u_{jk}} \nonumber \\&= \,\text {tr}\,\biggl (\Bigl (\frac{\partial \varepsilon _{a}^\mathrm{el-pl}(q_{lF})}{\partial q_{lF}} \Bigr )^T\frac{\partial q_{lF}}{\partial u_{jk}}\biggr ), \end{aligned}$$

(37)

where for $s,n=1,2,3$,

$$\begin{aligned}&\frac{\partial (q_{lF})_{sn}}{\partial u_{kj}} \nonumber \\&\quad =\frac{\partial }{\partial u_{jk}}\left( \sum _{m=1}^{n_\mathrm{sh}}u_{{\widehat{j}}(K(F),m)s}\left( ({\widehat{\nabla }}T_{K(F)}^{T} ({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_m({\widehat{\xi }}_l^F)\right) _n\right) \nonumber \\&\quad =\sum _{m=1}^{n_{\mathrm{sh}}}\frac{\partial u_{{\widehat{j}}(K(F),m)s}}{\partial u_{jk}}\left( ({\widehat{\nabla }}T_{K(F)}^{T} ({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_m({\widehat{\xi }}_l^F)\right) _n \nonumber \\&\quad ={\left\{ \begin{array}{ll} \delta _{sk}\left( ({\widehat{\nabla }}T_{K(F)}^{T} ({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_{{\widehat{j}}^{-1}_{K(F)}(j)} ({\widehat{\xi }}_l^F)\right) _n&{} { \text{ if } j\in {\widehat{j}}_{K(F)}}\\ 0&{} \text{ otherwise }\end{array}\right. }, \end{aligned}$$

(38)

with $\delta _{sk}$ Kronecker’s symbol and ${\widehat{j}}_{K(F)} = {\widehat{j}}_{K(F)}(\{1,\ldots ,n_{\mathrm{sh}}\})$. In order to simplify the calculation, we square (5) and divide by E. The resulting relation between $\varepsilon _{a}^{\mathrm{el-pl}}$ and $\frac{\sigma _a^2}{E}$ then is denoted by $\varepsilon _{a}^\mathrm{el-pl}=\bar{\mathrm{SD}}\left( \frac{\sigma _a^2}{E}\right)$. The partial derivative of $\varepsilon _{a}^{\mathrm{el-pl}}$ with respect to $q_{sm}$ is now:

$$\begin{aligned} \frac{\partial \varepsilon _{a}^{\mathrm{el-pl}}}{\partial q_{sm}}=\frac{d \mathrm{RO}\circ \bar{\mathrm{SD}}^{-1}\left( \frac{(\sigma _a)^2}{E}\right) }{d \left( \frac{(\sigma _a)^2}{E}\right) }\cdot \frac{\partial }{\partial q_{sm}}\biggl (\frac{(\sigma _a)^2}{E}\biggr ). \end{aligned}$$

(39)

The derivative of the function $\mathrm{RO}\circ \bar{\mathrm{SD}}^{-1}(\cdot )$ is easily calculated from (3) and (5). We further calculate

$$\begin{aligned} \frac{\partial }{\partial q_{sm}} \biggl (\frac{(\sigma _a)^2}{E}\biggr ) = \frac{3\mu ^2}{4E}\sum _{i,j=1}^{3}A_{ij}\frac{\partial A_{ij}}{\partial q_{sm}} \end{aligned}$$

(40)

using $A_{ij}=\biggl (-\frac{2}{3}\bigl (\frac{1}{2}(q_{ij}+q_{ji})\bigr )\delta _{ij}+q_{ij}+q_{ji}\biggr )$. We thus obtain

$$\begin{aligned} \frac{\partial A_{ij}}{\partial q_{s,m}}=-\frac{2}{3} \delta _{ij}\delta _{sm}+ \delta _{is}\delta _{jm}+\delta _{js}\delta _{im}. \end{aligned}$$

(41)

We thus get for the left hand side of (40)

$$\begin{aligned}&\frac{3\mu ^2}{4E}\sum _{i,j=1}^{3}\biggl (-\frac{2}{3} \bigl (\frac{1}{2}(q_{ij}+q_{ji})\bigr )\delta _{ij}+q_{ij}+q_{ji}\biggr ) \nonumber \\&\quad \bigl (-\frac{2}{3}\delta _{ij}\delta _{sm}+\delta _{is}\delta _{jm}+ \delta _{js}\delta _{im}\bigr ). \end{aligned}$$

(42)

1.4 Computing $\frac{\partial J}{\partial X}$

As in Sect. A.3, We use the abbreviation $q_{lF}$ for $\nabla u(\xi _l^F)$, see (34). The partial derivative of J(X, U) w.r.t. the global jth mesh node i-coordinate $X_{ji}$, $i=1,2,3$ and $j=1,\ldots ,N$, is

$$\begin{aligned}&\frac{\partial J(X,U)}{\partial X_{ji}} \nonumber \\&\quad =\sum _{\begin{array}{c} F\in {\mathcal {N}}_h \\ j\in {\widehat{j}}_{K(F)}(\{1,\ldots ,n_\mathrm{sh}\}) \end{array}}\sum _{l=1}^{l_q^F}\frac{\partial }{\partial X_{ji}}\Biggl (\omega _{lF}\left( \mathrm{CMB}^{-1}\left( \varepsilon _{a}^\mathrm{el-pl}(q_{lF})\right) \right) ^{-{\bar{m}}}\Biggr ) \nonumber \\&\quad =\sum _{\begin{array}{c} F\in {\mathcal {N}}_h \\ j\in {\widehat{j}}_{K(F)}(\{1,\ldots ,n_\mathrm{sh}\}) \end{array}}\sum _{l=1}^{l^F_q}\Biggl [\frac{\partial \omega _{lF}}{\partial X_{ji}}\Bigl (\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF})))\Bigr )^{-{\bar{m}}} \nonumber \\&\qquad +\, \omega _{lF}\frac{\partial }{\partial X_{ji}}\Bigl (\Bigl (\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF}))\Bigr )^{-{\bar{m}}}\Bigr )\Biggr ]. \end{aligned}$$

(43)

We compute first the partial derivative of $\frac{\partial \omega _{lF}}{\partial X_{ji}}$ as

$$\begin{aligned} \frac{\partial \omega _{lF}}{\partial X_{ji}}&= \frac{\partial }{\partial X_{ji}}\left( \widehat{\omega _l} \sqrt{\det (g_F({\widehat{\xi }}_l^F))}\right) \nonumber \\&=\widehat{\omega _l}\frac{1}{2}\bigl (\det (g_F({\widehat{\xi }}_l^F)) \bigr )^{-1/2}\frac{\partial }{\partial X_{ji}}(\det (g_F({\widehat{\xi }}^F_l))). \end{aligned}$$

(44)

The derivative of the determinant is

$$\begin{aligned}&\frac{\partial }{\partial X_{ji}}\biggl (\det \Bigl (g_F({\widehat{\xi }}_l^F)\Bigr )\biggr ) \nonumber \\&\quad =\det \Bigl (g_F({\widehat{\xi }}^F_l)\Bigr ) \,\text {tr}\,\biggl (\Bigl (g_F({\widehat{\xi }}^F_l)\Bigr )^{-1}\frac{\partial g_F({\widehat{\xi }}^F_l)}{\partial X_{ji}}\biggr ), \end{aligned}$$

(45)

where $g_F({\widehat{\xi }}_l^F)=J_F({\widehat{\xi }}_l^F)^T J_F({\widehat{\xi }}_l^F)$ and

$$\begin{aligned} J_F({\widehat{\xi }}_l^F)&=\frac{\partial T_K({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}^F} \nonumber \\&=\sum _{r=1}^{n_{\mathrm{sh}}} \left( \begin{array}{c@{\quad }c} X_{r1}^{K(F)}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}_1^F} &{} X_{1r}^{K(F)}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}_2^F} \\ X_{r2}^{K(F)}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}_1^F} &{} X_{r2}^{K(F)}\frac{\partial \theta _r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}^F_2} \\ X_{r3}^{K(F)}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}_1^F} &{} X_{r3}^{K(F)}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial {\widehat{X}}^F_2} \\ \end{array} \right) . \end{aligned}$$

(46)

Here ${\widehat{X}}^F_{i}$, $i=1,2$, are the coordinates on the two dimensional reference face ${\widehat{F}}$ corresponding to F in ${\widehat{K}}$.

The derivative $\frac{\partial g_F({\widehat{\xi }}_l^F)}{\partial X_{ji}}$ is thus given by

$$\begin{aligned} \frac{\partial g_F({\widehat{\xi }}_l^F)}{\partial X_{ji}}=&\frac{\partial }{\partial X_{ji}}\bigl (J_F({\widehat{\xi }}_l^F)\bigr )^T J_F({\widehat{\xi }}_l^F) \nonumber \\&+\bigl (J_F({\widehat{\xi }}_l^F)\bigr )^T\frac{\partial }{\partial X_{ji}}\bigl (J_F({\widehat{\xi }}_l^F)\bigr ). \end{aligned}$$

(47)

Furthermore, for $s=1,2,3$ and $k=1,2$ we have

$$\begin{aligned}&\frac{\partial }{\partial X_{ji}}\bigl (J_F({\widehat{\xi }}_l^F)_{sk}\bigr )= \sum _{r=1}^{n_\mathrm{sh}}\frac{\partial }{\partial X_{ji}}\left( X_{rs}^{K(F)}\frac{\partial \theta _r({\widehat{\xi }}_l^F)}{\partial \widehat{X_k}} \right) \nonumber \\&\quad ={\left\{ \begin{array}{ll} \delta _{is}\frac{\partial {\widehat{\theta }}_{{\widehat{j}}^{-1}_{K(F)}(j)}({\widehat{\xi }}_l^F)}{\partial \widehat{X_k}}&{}\quad \text{ if } j\in \widehat{j}_{K(F)}(\{1,\ldots ,n_{\mathrm{sh}}\})\\ 0&{}\quad \text{ otherwise } \end{array}\right. }. \end{aligned}$$

(48)

This finishes the computation of the first term on the right hand side of (43).

To compute the second term in (43), we take the partial derivative

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial X_{ji}}\Bigl [\mathrm{CMB}^{-1}(\varepsilon _{a}^{\mathrm{el-pl}}(q_{lF}))\Bigr ]^{-{\bar{m}}} \\&\quad =\frac{-{\bar{m}}[\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF}))]^{-{\bar{m}}-1}}{\frac{\partial \mathrm{CMB}}{\partial \varepsilon _{a}^{\mathrm{el-pl}}}(\mathrm{CMB}^{-1}(\varepsilon _{a}^\mathrm{el-pl}(q_{lF})))} \frac{\partial }{\partial X_{ji}}\bigl (\varepsilon _{a}^{\mathrm{el-pl}}(q_{lF})\bigr ), \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \frac{\partial }{\partial X_{ji}}\bigl (\varepsilon _{a}^\mathrm{el-pl}(q_{lF})\bigr )&= \frac{\partial \varepsilon _{a}^\mathrm{el-pl}(q_{lF})}{\partial q_{lF}}:\frac{\partial q_{lF}}{\partial X_{ji}}. \end{aligned}$$

(49)

In the above equation,  :  stands for the contraction of both q indices. Next we have to compute

$$\begin{aligned}&\frac{\partial q_{lF}}{\partial X_{ji}}(X) \nonumber \\&\quad =\frac{\partial }{\partial X_{ji}}\biggl (\sum _{m=1}^{n_\mathrm{sh}}u_{{\widehat{j}}(K(F),m)}\otimes (({\widehat{\nabla }}T_{K(F)}^T ({\widehat{\xi }}_l^F))^{-1}{\widehat{\nabla }} {\widehat{\theta }}_m({\widehat{\xi }}_l^F))\biggr ) \nonumber \\&\quad =\sum _{m=1}^{n_\mathrm{sh}}u_{j(K(F),m)}\otimes \Bigl (\frac{\partial }{\partial X_{ji}}({\widehat{\nabla }}T_{K(F)}^T({\widehat{\xi }}_l^F))^{-1} \Bigr ){\widehat{\nabla }} {\widehat{\theta }}_m({\widehat{\xi }}_l^F), \end{aligned}$$

(50)

where

$$\begin{aligned}&\frac{\partial }{\partial X_{ji}}\bigl ({\widehat{\nabla }} T_{K}^{T}({\widehat{\xi }}_l^ F)\bigr )^{-1} \nonumber \\&\quad = -\bigl ({\widehat{\nabla }} T_{K}^T({\widehat{\xi }}_l^ F)\bigr )^{-1}\frac{\partial \bigl ({\widehat{\nabla }} T_{K}^T({\widehat{\xi }}_l^ F)\bigr )}{\partial X_{ji}}\bigl ({\widehat{\nabla }} T_{K}^T({\widehat{\xi }}_l^ F)\bigr )^{-1}. \end{aligned}$$

(51)

The Jacobian matrix has the form

$$\begin{aligned} {\widehat{\nabla }} T_{K}({\widehat{\xi }}_l^ F)&= \left( \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial X_{1}^{K}}{\partial \widehat{X_1}} &{} \frac{\partial X_{1}^{K}}{\partial \widehat{X_2}} &{} \frac{\partial X_{1}^{K}}{\partial \widehat{X_3}} \\ \frac{\partial X_{2}^{K}}{\partial \widehat{X_1}} &{} \frac{\partial X_{2}^{K}}{\partial \widehat{X_2}} &{} \frac{\partial X_{2}^{K}}{\partial \widehat{X_3}}\\ \frac{\partial X_{3}^{K}}{\partial \widehat{X_1}} &{} \frac{\partial X_{3}^{K}}{\partial \widehat{X_2}} &{} \frac{\partial X_{3}^{K}}{\partial \widehat{X_3}}\\ \end{array} \right) \nonumber \\&=\sum _{r=1}^{n_{\mathrm{sh}}} \left( \begin{array}{c@{\quad }c@{\quad }c} X_{r1}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_1}} &{} X_{r1}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_2}} &{} X_{r1}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_3}} \\ X_{r2}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_1}} &{} X_{r2}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_2}} &{} X_{r2}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_3}}\\ X_{r3}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_1}} &{} X_{r3}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_2}} &{} X_{r3}^{K}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^ F)}{\partial \widehat{X_3}}\\ \end{array} \right) . \end{aligned}$$

(52)

Finally, we get

$$\begin{aligned}&\frac{\partial \bigl ({\widehat{\nabla }} T_{K}^T({\widehat{\xi }}_l^ F)_{sk}\bigr )}{\partial X_{ji}}=\sum _{r=1}^{n_\mathrm{sh}}\frac{\partial X_{rk}^{K}}{\partial X_{ji}}\frac{\partial {\widehat{\theta }}_r({\widehat{\xi }}_l^F)}{\partial \widehat{X_s}} \nonumber \\&\quad ={\left\{ \begin{array}{ll}\delta _{ik} \frac{\partial {\widehat{\theta }}_{{\widehat{j}}^{-1}_{K}(j)}({\widehat{\xi }}_l^ F)}{\partial \widehat{X_s}},&{} \text{ if } j\in {\widehat{j}}_{K}(\{1,\ldots ,n_{\mathrm{sh}}\})\\ 0&{} \text{ otherwise }\end{array}\right. }. \end{aligned}$$

(53)

This finishes the computation of the second term.

1.5 Computing $\frac{\partial B}{\partial X}$

As in (30) we have

$$\begin{aligned} B(X)_{(q,r),(k,s)}&=B(e_r\theta _q,e_s\theta _k) \nonumber \\&=B_1(e_r\theta _q,e_s\theta _k)+B_2(e_r\theta _q,e_s\theta _k), \end{aligned}$$

then $\frac{\partial B(X)_{(q,r),(k,s)}}{\partial X}=\frac{\partial B_1(e_r\theta _q,e_s\theta _k)}{\partial X_{ji}}+\frac{\partial B_2(e_r\theta _q,e_s\theta _k)}{\partial X_{ji}}$. For the first partial derivative, one obtains

$$\begin{aligned} \frac{\partial B_1(e_r\theta _q,e_s\theta _k)}{\partial X_{ji}}&=\lambda \sum _{K\in {\widehat{j}}^{-1}(j)_1\cap {\widehat{j}}^{-1}(q)_1\cap {\widehat{j}}^{-1}(k)_1}\sum _{l=1}^{l_q} \nonumber \\&\Bigg \{\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr )\nabla _r \theta _q(\xi _l^{K})\nabla _s \theta _k(\xi _l^{K}) \nonumber \\&\quad +\omega _{lK}\frac{\partial }{\partial X_{ji}}\bigl (\nabla _r \theta _q(\xi _l^{K})\bigr )\nabla _s \theta _k(\xi _l^{K}) \nonumber \\&\quad +\omega _{lK}\nabla _r \theta _q(\xi _l^{K})\frac{\partial }{\partial X_{ji}}\bigl (\nabla _s \theta _k(\xi _l^{K})\bigr )\Bigg \}. \end{aligned}$$

(54)

Here we use the convention that ${\widehat{j}}^{-1}(j)_1$ is the projection of the set ${\widehat{j}}^{-1}(j)$ to the first component (the index of the element).

We thus have to compute the three partial derivatives $\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr )$, $\frac{\partial }{\partial X_{ji}}\bigl (\nabla _r\theta _q(\xi _l^K)\bigr )$ and $\frac{\partial }{\partial X_{ji}}\bigl (\nabla _s\theta _k(\xi _l^K)\bigr )$. We start with the first partial derivative and we obtain

$$\begin{aligned} \frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr )&=\frac{\partial }{\partial X_{ji}}\bigl (\widehat{\omega _{l}}\det ({\widehat{\nabla }} T_K(\widehat{\xi _l})\bigr ) \nonumber \\&=\widehat{\omega _l}\frac{\partial }{\partial X_{ji}}\bigl (\det ( {\widehat{\nabla }}T_K(\widehat{\xi _l})\bigr ). \end{aligned}$$

(55)

For an invertible matrix A, we have

$$\begin{aligned} \frac{d}{d\alpha }\det (A)=\det (A)\,\text {tr}\,(A^{-1}\frac{dA}{d\alpha }), \end{aligned}$$

so we get

$$\begin{aligned}&\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr ) \nonumber \\&\quad =\widehat{\omega _l}\det \left( {\widehat{\nabla }} T_K(\widehat{\xi _l})\right) \,\text {tr}\,\left( ({\widehat{\nabla }}T_K({\widehat{\xi }}_l))^{-1}\frac{\partial {\widehat{\nabla }}T_K({\widehat{\xi }}_l)}{\partial X_{ji}}\right) . \end{aligned}$$

(56)

The partial $X_{ji}$ derivative of the matrix ${\widehat{\nabla }}T_K({\widehat{\xi }}_l)$ has been calculated in (53), where we have to replace ${\widehat{\xi }}_l^F$ with ${\widehat{\xi }}_l$.

The second partial derivative in $\frac{\partial B_1}{\partial X_{ji}}$ is easily calculated

$$\begin{aligned} \frac{\partial \nabla _r \theta _q}{\partial X_{ji}}(\xi _l^K)&= \frac{\partial }{\partial X_{ji}}\left[ \left( {\widehat{\nabla }}T_K({\hat{\xi }}_l)^T \right) ^{-1}{\widehat{\nabla }} {\widehat{\theta }}_{{\widehat{j}}_{K}^{-1}(q)}({\hat{\xi }}_l)\right] _r. \end{aligned}$$

(57)

We now refer to Eqs. (51)–(53) in in Sect. A.4 to conclude the computation. Here, of course, the surface quadrature point ${\hat{\xi }}^F_l$ has to be replaced by the volume quadrature point ${\hat{\xi }}_l$. The third partial derivative in (54) is completely analogous to the second.

For the partial derivative $\frac{\partial B_2}{\partial X}$ we obtain

$$\begin{aligned}&\frac{\partial B_2(e_r\theta _q,e_s\theta _k)}{\partial X_{ji}} \nonumber \\&\quad =2\mu \sum _{K\in {\widehat{j}}^{-1}(j)_1\cap {\widehat{j}}^{-1}(q)_1\cap {\widehat{j}}^{-1}(k)_1}\nonumber \\&\qquad \sum _{l=1}^{l_q} \Bigg \{\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr )\varepsilon \left( e_r \theta _q(\xi _l^{K}))\right) :\varepsilon \left( e_s \theta _k(\xi _l^{K})\right) \nonumber \\&\quad \qquad +\omega _{lK}\frac{\partial }{\partial X_{ji}}\bigl (\varepsilon \left( e_r \theta _q(\xi _l^{K}))\right) \bigr ):\varepsilon \left( e_s \theta _k(\xi _l^{K})\right) \nonumber \\&\quad \qquad +\omega _{lK}\varepsilon \left( e_r \theta _q(\xi _l^{K}))\right) :\frac{\partial }{\partial X_{ji}}\bigl (\varepsilon \left( e_s \theta _k(\xi _l^{K})\right) \bigr )\Bigg \}. \end{aligned}$$

(58)

The first term is calculated in (55). For the second term, we observe that the linear elastic strain tensor field is given by

$$\begin{aligned} \varepsilon (e_r\theta _q(\xi _l^ K))_{ab}= \frac{1}{2}\left( \delta _{rb}\nabla _a \theta _q(\xi _l^ K))+\delta _{ra}\nabla _b \theta _q(\xi _l^ K)\right) ,\nonumber \\ \end{aligned}$$

(59)

and we refer to the argument following Eq. (57) to conclude the computation of (58).

1.6 Computing $\frac{\partial F}{\partial X}$

We have $\frac{\partial F_{(q,r)}}{\partial X_{ji}}=\frac{\partial F_{(q,r)}^{\mathrm{vol}}}{\partial X_{ji}}+\frac{\partial F_{(q,r)}^\mathrm{surf}}{\partial X_{ji}}$. Starting with the volume term, we obtain

$$\begin{aligned} \frac{\partial F_{(q,r)}^{\mathrm{vol}}}{\partial X_{ji}}&=\sum _{K\in {\widehat{j}}^{-1}(j)_1\cap {\widehat{j}}^{-1}(q)_1} \nonumber \\&\quad \sum _{l=1}^{l_q}\Bigg \{\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lK}\bigr )f_r(\xi _l^K)\theta _q(\xi _l^K) \nonumber \\&\quad \qquad +\,\omega _{lK}\frac{\partial }{\partial X_{ji}}\bigl (f_r(\xi _l^K)\bigr )\theta _q(\xi _l^K) \nonumber \\&\quad \qquad +\,\omega _{lK}f_r(\xi _l^K)\frac{\partial }{\partial X_{ji}}\bigl (\theta _q(\xi _l^K)\bigr )\bigg \} \end{aligned}$$

(60)

The partial derivative of the volume weight has been calculated in (55). The third term in (60) vanishes, as $\theta _q(\theta _l)={\hat{\theta }}_{{\widehat{j}}^{-1}_K(q)}({\hat{\xi }}_l)$ does not depend on $X_{ji}$. Unless the volume force density f does depend explicitly on the position (like in the case of centrifugal loads) this term vanishes as the third one.

Finally we have to calculate the partial derivative of the surface loads

$$\begin{aligned} \frac{\partial F_{(q,r)}^{\mathrm{surf}}}{\partial X_{ji}}&= \sum _{\begin{array}{c} F\in {\mathcal {N}}_h \\ K(F)\in {\widehat{j}}^{-1}(j)_1 \cap {\widehat{j}}^{-1} (q)_1 \end{array}} \nonumber \\&\quad \sum _{l=1}^{l_q^F}\Bigg \{\frac{\partial }{\partial X_{ji}}\bigl (\omega _{lF}\bigr )f_r(\xi _l^F)\theta _q(\xi _l^F) \nonumber \\&\quad \qquad +\,\omega _{lF}\frac{\partial }{\partial X_{ji}}\bigl (f_r(\xi _l^F)\bigr )\theta _q(\xi _l^F) \nonumber \\&\quad \qquad +\,\omega _{lF}f_r(\xi _l^F)\frac{\partial }{\partial X_{ji}}\bigl (\theta _q(\xi _l^F)\bigr )\Bigg \}. \end{aligned}$$

(61)

The first term can be calculated with the aid of (44). For the second and third term, the same reasoning applies as for partial derivative of the volume force.