Affine deformations serve as basic examples in the continuum mechanics of deformable three-dimensional bodies (usually referred to as homogeneous deformations). They preserve parallelism of straight lines, and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time-dependent deformations (motions) of such a continuum, but in a more general context, by considering that it is modeled by a Riemannian hypersurface. First we prove certain equivalent formulas for the variation of the connection of the hypersurface. Some of these formulas are expressed in terms of geometrical quantities, and others in terms of kinematical quantities of the deforming continuum. The latter is achieved by using an adapted version of the polar decomposition theorem, frequently used in continuum mechanics to analyze motion. We also apply our results to special motions like tangential and normal motions. Further, we find necessary and sufficient conditions for this variation to be zero (infinitesimal affine motions), providing insight on the form of these motions and the kind of hypersurfaces that allow such motions. Finally, we give some specific examples of mechanical interest which demonstrate motions that are infinitesimally affine but not infinitesimally isometric.
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This is usually achieved via a geometric parameterization of its boundary via level-set functions. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form −∆u + f1(u) = f2 with polynomial nonlinearity via the cut finite element method. Boundary conditions are enforced, using a Nitsche-type approach. To ensure stability and error estimates that are independent of the position of the boundary with respect to the mesh, the formulations are augmented with additional boundary zone ghost penalty terms. These terms act on the jumps of the normal gradients at faces associated with cut elements. A-priori error estimates are derived, while numerical examples illustrate the implementation of the method and validate the theoretical findings.
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