On shape sensitivities with heaviside-enriched XFEM

Abstract

This paper investigates the behavior of shape sensitivities within the context of the eXtended Finite Element Method (XFEM) using a Heaviside enrichment strategy, wherein the shape derivative is computed by the adjoint method. The Heaviside function is discontinuous by construction. This feature of the enrichment function presents advantages as well as challenges in the computation of shape sensitivities, both of which are discussed in detail in this paper. Using continuum and discrete approaches, we present the derivation of analytical shape sensitivities with respect to the design variables which define the design geometry. We propose a robust semi-analytical approach to computing the shape sensitivities, which provides great ease of implementation as compared to fully analytical approaches. The behavior of the XFEM-based shape sensitivities is analyzed using linear heat diffusion examples in 2D, and an incompressible fluid flow example in 3D. We compare XFEM-based shape sensitivities against shape sensitivities obtained through the classical approach of using a body-fitted mesh. It is found that the former are not as smooth as those obtained using a comparable body-fitted mesh. This discrepancy is shown to be an outcome of the discretization error of the design geometry on a background mesh and is not a consequence of the approach by which the XFEM-based shape sensitivities are computed.

Appendix: A

CAD-free based shape optimization studies use the nodal coordinates of the element as the design variables. Here we draw a quantitative comparison between shape sensitivities obtained using the approach proposed in the current study (Eulerian framework), and shape sensitivities obtained using a body-fitted mesh (Lagrangian framework).

We consider a 1D linear heat diffusion problem depicted in Fig. 26. A unit heat flux is applied to node 3 and to the interface x _Γ, in the body-fitted and XFEM cases respectively. For the XFEM case, phase 2 is void of any material. Consequently, the setups shown in Fig. 26 are physically equivalent. The response, $\mathcal {Z}$ is the temperature measured at node 2. As discussed in Section 4.2, shape sensitivities vanish for elements not intersected by the material interface. Thus, we draw comparisons for the second element only, the system of equations for which can be expressed as

Fig. 26

Heat diffusion problem setup in 1D

$$\begin{array}{@{}rcl@{}} &&\text{FEM:} \\ &&\left[\begin{array}{cc} \frac{1}{L_{1}} + \frac{1}{L_{2}} & -\frac{1}{L_{2}} \\ -\frac{1}{L_{2}} & \frac{1}{L_{2}} \end{array}\right] \left[\begin{array}{c} \hat{u}_{2} \\ \hat{u}_{3} \end{array}\right] = \left[\begin{array}{c} 0 \\ 1 \end{array}\right] \implies \\ &&\left[\begin{array}{cc} \frac{1}{x_{2} - x_{1}} + \frac{1}{x_{3} - x_{2}} & -\frac{1}{x_{3} - x_{2}} \\ -\frac{1}{x_{3} - x_{2}} & \frac{1}{x_{3} - x_{2}} \end{array}\right] \left[\begin{array}{c} \hat{u}_{2} \\ \hat{u}_{3} \end{array}\right] = \left[\begin{array}{c} 0 \\ 1 \end{array}\right],\\ &&\text{XFEM:} \\ &&\left[\begin{array}{cc} \frac{1}{L_{1}} + \frac{L_{2}}{{L_{0}^{2}}} & -\frac{L_{2}}{{L_{0}^{2}}} \\ - \frac{L_{2}}{{L_{0}^{2}}} & \frac{L_{2}}{{L_{0}^{2}}} \end{array}\right] \left[\begin{array}{c} \hat{u}_{2}^{*} \\ \hat{u}_{3}^{*} \end{array}\right] = \left[\begin{array}{c} \frac{L_{0} - L_{2}}{L_{0}} \\ \frac{L_{2}}{L_{0}} \end{array}\right] \implies \\ &&\left[\begin{array}{cc} \frac{1}{x_{2} - x_{1}} + \frac{(x_{\Gamma} - x_{2})}{(x_{3} - x_{2})^{2}} & -\frac{x_{\Gamma} -x_{2}}{(x_{3} - x_{2})^{2}} \\ -\frac{x_{\Gamma} -x_{2}}{(x_{3} - x_{2})^{2}} & \frac{x_{\Gamma} -x_{2}}{(x_{3} - x_{2})^{2}} \end{array}\right] \left[\begin{array}{c} \hat{u}_{2}^{*} \\ \hat{u}_{3}^{*} \end{array}\right] = \left[\begin{array}{c} \frac{x_{3} - x_{\Gamma}}{x_{3} - x_{2}} \\ \frac{x_{\Gamma} - x_{2}}{x_{3} - x_{2}} \end{array}\right], \end{array} $$

(50)

where the force vector for the XFEM was obtained using the residual contribution from the Neumann boundary condition in (10). The discretized solution using FEM and XFEM are denoted by $\hat {u}$ and $\hat {u}^{*}$ respectively. Note, it is trivial to show following a few calculations that $\hat {u}^{*}(x_{\Gamma }) = \hat {u}_{3} = L_{1} + L_{2}$ and ${d\hat {u}^{*}(x_{\Gamma })}/{dx_{\Gamma }} = {d\hat {u}_{3}}/{dx_{3}} = 1$, thus proving the equivalence of the two approaches.

To compute the shape sensitivities using the adjoint approach (Section 4), we require the derivatives of the response function with respect to the solution, and derivative of the residual with respect to the solution and the material interface. Following (50), the residual of the two systems is written as

$$\begin{array}{@{}rcl@{}} &&\!\!\!\!\text{FEM:} \\ &&\!\!\!\! \boldsymbol{\mathcal{R}} = \left[\begin{array}{c} \hat{u}_{2}\frac{1}{x_{2} - x_{1}} + \hat{u}_{2}\frac{1}{x_{3} - x_{2}} - \hat{u}_{3}\frac{1}{x_{3} - x_{2}} \\ -\hat{u}_{2}\frac{1}{x_{3} - x_{2}} + \hat{u}_{3}\frac{1}{x_{3} - x_{2}} - 1 \end{array}\right] , \\ &&\!\!\!\!\text{XFEM:} \\ &&\!\!\!\! \boldsymbol{\mathcal{R}} \,=\, \left[\begin{array}{c} \hat{u}_{2}^{*}\frac{1}{x_{2} - x_{1}} + \hat{u}_{2}^{*}\frac{x_{\Gamma} - x_{2}}{(x_{3} - x_{2})^{2}} - \hat{u}_{3}^{*}\frac{x_{\Gamma} - x_{2}}{(x_{3} - x_{2})^{2}} - \frac{x_{3} - x_{\Gamma}}{x_{3} - x_{2}} \\ - \hat{u}_{2}^{*}\frac{x_{\Gamma} - x_{2}}{(x_{3} - x_{2})^{2}} + \hat{u}_{3}^{*}\frac{x_{\Gamma} - x_{2}}{(x_{3} - x_{2})^{2}} - \frac{x_{\Gamma} - x_{2}}{x_{3} - x_{2}} \end{array}\!\right] . \end{array} $$

(51)

The residual derivatives with respect to the material interface are then given by

$$\begin{array}{@{}rcl@{}} &&\text{FEM:} \\ &&\frac{\partial \boldsymbol{\mathcal{R}}}{\partial x_{3}}= \left[\begin{array}{c} -\hat{u}_{2}\frac{1}{(x_{3} - x_{2})^{2}} + \hat{u}_{3}\frac{1}{(x_{3} - x_{2})^{2}} \\ \hat{u}_{2}\frac{1}{(x_{3} - x_{2})^{2}} - \hat{u}_{3}\frac{1}{(x_{3} - x_{2})^{2}} \end{array}\right] , \\ &&\text{XFEM:} \\ &&\frac{\partial \boldsymbol{\mathcal{R}}}{\partial x_{\Gamma}}= \left[\begin{array}{c} \hat{u}_{2}^{*} \frac{1}{(x_{3} - x_{2})^{2}} - \hat{u}_{3}^{*}\frac{1}{(x_{3} - x_{2})^{2}} + \frac{1}{x_{3} - x_{2}} \\ - \hat{u}_{2}^{*}\frac{1}{(x_{3} - x_{2})^{2}} + \hat{u}_{3}^{*}\frac{1}{(x_{3} - x_{2})^{2}} - \frac{1}{x_{3} - x_{2}} \end{array}\right] . \end{array} $$

(52)

The residual derivatives with respect to the solution are given by the left hand side matrices in (50). The derivative of the response function with respect to the solution is same for the two approaches, and is given by,

$$ \frac{\partial \mathcal{Z}}{\partial \hat{\textit{\textbf{u}}}^{*}} = \frac{\partial \mathcal{Z}}{\partial \hat{\textit{\textbf{u}}}} = \left[\begin{array}{c} 1 \\ 0 \end{array}\right]. $$

(53)

The vector of adjoint variables is computed using (23) to give

$$\begin{array}{@{}rcl@{}} \boldsymbol{\lambda} &=& -\left( \frac{\partial \boldsymbol{\mathcal{R}}}{\partial \hat{\textit{\textbf{u}}}} \right)^{-1} \frac{\partial \mathcal{Z}}{\partial \hat{\textit{\textbf{u}}}} = \left[\begin{array}{c} x_{1} - x_{2} \\ x_{1} - x_{2} \end{array}\right] , \\ \boldsymbol{\lambda}^{*} &=& -\left( \frac{\partial \boldsymbol{\mathcal{R}}}{\partial \hat{\textit{\textbf{u}}}^{*}} \right)^{-1} \frac{\partial \mathcal{Z}}{\partial \hat{\textit{\textbf{u}}}^{*}} = \left[\begin{array}{c} x_{1} - x_{2} \\ x_{1} - x_{2} \end{array}\right] , \end{array} $$

(54)

where λ and λ ^∗ denote the vector of adjoint variables for the FEM case and the XFEM case respectively. The shape sensitivities are readily obtained following a dot product between (54) and (52). Upon comparison of the sensitivities obtained using the two frameworks, a sign change in the sensitivities resulting from (52) is evident, along with the presence of an additional term in the XFEM case. This difference is the result of extrapolating the force vector from the material interface on to the finite element nodes.