SUMMARYWe adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low-order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses postprocessing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous-Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous-Galerkin-based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427-454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher-order geometric primitives. Secondly, we employ higherorder shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re-entrant corner. Results are compared to a low-order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations.
SUMMARYWe present an efficient numerical method to solve for cyclic steady states of nonlinear electro-mechanical devices excited at resonance. Many electro-mechanical systems are designed to operate at resonance, where the ramp-up simulation to steady state is computationally very expensive -especially when low damping is present. The proposed method relies on a Newton-Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time-stepping from zero initial conditions. We use a recently developed high-order Eulerian-Lagrangian finite element method in combination with an energy-preserving dynamic contact algorithm in order to solve the coupled electro-mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro-electro-mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed-ups of the form S D 0:6 0:8 , where is the linear damping ratio of the corresponding resonance frequency.
Existing analytical and numerical methodologies are discussed and then extended in order to calculate critical contamination-particle sizes, which will result in deleterious effects during EUVL E-chucking in the face of an error budget on the image-placement-error (IPE). The enhanced analytical models include a gap dependant clamping pressure formulation, the consideration of a general material law for realistic particle crushing and the influence of frictional contact. We present a discussion of the defects of the classical de-coupled modeling approach where particle crushing and mask/chuck indentation are separated from the global computation of mask bending. To repair this defect we present a new analytic approach based on an exact Hankel transform method which allows a fully coupled solution. This will capture the contribution of the mask indentation to the image-placement-error (estimated IPE increase of 20%). A fully coupled finite element model is used to validate the analytical models and to further investigate the impact of a mask back-side CrN-layer. The models are applied to existing experimental data with good agreement. For a standard material combination, a given IPE tolerance of 1 nm and a 15 kPa closing pressure, we derive bounds for single particles of cylindrical shape (radius × height < 44 μm) and spherical shape (diameter < 12 μm).
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