A note on the stability parameter in Nitsche's method for unfitted boundary value problems

Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. Of the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors. <br/

On the effects of mechanical stress of biological membranes in modeling of swelling dynamics of biological systems

We highlight mechanical stretching and bending of membranes and the importance of membrane deformations in the analysis of swelling dynamics of biological systems, including cells and subcellular organelles. Membrane deformation upon swelling generates tensile stress and internal pressure, contributing to volume changes in biological systems. Therefore, in addition to physical (internal/external) and chemical factors, mechanical properties of the membranes should be considered in modeling analysis of cellular swelling. Here we describe an approach that considers mechanical properties of the membranes in the analysis of swelling dynamics of biological systems. This approach includes membrane bending and stretching deformations into the model, producing a more realistic description of swelling. We also discuss the effects of membrane stretching on swelling dynamics. We report that additional pressure generated by membrane bending is negligible, compared to pressures generated by membrane stretching, when both membrane surface area and volume are variable parameters. Note that bending deformations are reversible, while stretching deformation may be irreversible, leading to membrane disruption when they exceed a certain threshold level. Therefore, bending deformations need only be considered in reversible physiological swelling, whereas stretching deformations should also be considered in pathological irreversible swelling. Thus, the currently proposed approach may be used to develop a detailed biophysical model describing the transition from physiological to pathological swelling mode.National Aeronautics & Space Administration (NASA):80NSSC19M0049; PR Space Grant (NASA):NNX15AI11Hinfo:eu-repo/semantics/publishedVersio

First Order Error Correction for Trimmed Quadrature in Isogeometric Analysis

International audienceIn this work, we develop a specialized quadrature rule for trimmed domains , where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a pre-defined base case. We then extend the classical approach of linear approximation of the trimming curve by adding an error correction term based on a Taylor expansion of the blending between the linearized implicit trimming curve and the original one. This approach leads to an accurate method which improves the convergence of the quadrature error by one order compared to piecewise linear approximation of the trimming curve. It is at the same time efficient, since essentially the computation of one extra one-dimensional integral on each trimmed cell is required. Finally, the method is easy to implement, since it only involves one additional line integral and refrains from any point inversion or optimization operations. The convergence is analyzed theoretically and numerical experiments confirm that the accuracy is improved without compromising the computational complexity

Integrating a Constraint Solver into a Real-Time Animation Environment

This article investigates the integration of an interactive constraint solver into an existing 2-D real-time animation environment in order to achieve a better observability, traceability, and stability of the individual graphical objects. We present two approaches for assigning constraints to the objects. The first approach assigns constraints to the objects when they are created keeping them stable during their entire life-time. The second approach dynamically changes constraints before the computation of each frame. The investigation is based on our practical experience with the complete visual programming language Pictorial Janus and the parallel constraint solver Parcon