This paper presents a state of the art report of our project, the main objectives of which are: -- Categorization and summary of the geometric concepts required in a functionally based modeling environment; -- Elaboration of a rich system of geometric operations closed on functionally represented objects; -- Treatment of multidimensional and particularly space-time objects in a uniform manner; 6

This paper is concerned with the implementation of variational arbitrary Lagrangian–Eulerian formulations, also known as variational r-adaption methods. These methods seek to minimize the energy function with respect to the finite-element mesh over the reference configuration of the body. We propose a solution strategy based on a viscous regularization of the configurational forces. This procedure eliminates the ill-posedness of the problem without changing its solutions, i.e. the minimizers of the regularized problems are also minimizers of the original functional. We also develop strategies for optimizing the triangulation, or mesh connectivity, and for allowing nodes to migrate in and out of the boundary of the domain. Selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density. Copyright � 2006 John Wiley & Sons, Ltd.

This paper is concerned with the formulation of a variational r-adaption method for finite-deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configurational forces for isoparametric elements and non-linear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi-infinite crack in linear and non-linear elastic bodies; and the optimization of the shape of elastic inclusions.