Force Computation for Dielectrics Using Shape Calculus

Abstract We are concerned with the numerical computation of electrostatic forces/torques in only piece-wise homogeneous materials using the boundary element method (BEM). Conventional force formulas based on the Maxwell stress tensor yield functionals that fail to be continuous on natural trace spaces. Thus their use in conjunction with BEM incurs slow convergence and low accuracy. We employ the remedy discovered in [P. Panchal and R. Hiptmair, Electrostatic force computation with boundary element methods, SMAI J. Comput. Math. 8 (2022), 49–74]. Motivated by the virtual work principle which is interpreted using techniques of shape calculus, and using the adjoint method from shape optimization, we derive stable interface-based force functionals suitable for use with BEM. This is done in the framework of single-trace direct boundary integral equations for second-order transmission problems. Numerical tests confirm the fast asymptotic convergence and superior accuracy of the new formulas for the computation of total forces and torques.