ALBERT---Software for scientific computations and applications.
Algebraic domain decomposition solver for linear elasticity
We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn’s constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.
An a posteriori error analysis for dynamic viscoelastic problems
In this paper, a dynamic viscoelastic problem is numerically studied. The variational problem is written in terms of the velocity field and it leads to a parabolic linear variational equation. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. An a priori error estimates result is recalled, from which the linear convergence is derived under suitable regularity conditions. Then, an a posteriori...
An a posteriori error analysis for dynamic viscoelastic problems
In this paper, a dynamic viscoelastic problem is numerically studied. The variational problem is written in terms of the velocity field and it leads to a parabolic linear variational equation. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. An a priori error estimates result is recalled, from which the linear convergence is derived under suitable regularity conditions. Then, an a posteriori error...
An adaptive finite element method for solving a double well problem describing crystalline microstructure
An adaptive finite element method for solving a double well problem describing crystalline microstructure
The minimization of nonconvex functionals naturally arises in materials sciences where deformation gradients in certain alloys exhibit microstructures. For example, minimizing sequences of the nonconvex Ericksen-James energy can be associated with deformations in martensitic materials that are observed in experiments[2,3]. — From the numerical point of view, classical conforming and nonconforming finite element discretizations have been observed to give minimizers with their quality being highly dependent...
An additive Schwarz method for mortar Morley finite element discretizations of 4th order elliptic problem in 2D.
An Algorithm to Compute a Sparse Basis of the Null Space.
An analysis of a contact problem for a cylindrical shell: A primary and dual formulation
In this paper the contact problem for a cylindrical shell and a stiff punch is studied. The existence and uniqueness of a solution is verified. The finite element method is discussed.
An analysis of the boundary layer in the 1D surface Cauchy–Born model
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
An analysis of the boundary layer in the 1D surface Cauchy–Born model∗
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
An Analysis of the p-Version of the Finite Element Method for Nearly Incompressible Materials. Uniformly Valid, Optimal Error Estimates
An approximate solution for boundary value problems in structural engineering and fluid mechanics.
An asymptotic approach to inverse scattering problems on weakly nonlinear elastic rods.
An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions.
An elasto-plastic contact problem
An energy-preserving Discrete Element Method for elastodynamics∗
We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical...
An energy-preserving Discrete Element Method for elastodynamics∗
We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical...
An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment
We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in...
An equilibrium finite element method in three-dimensional elasticity
The tetrahedral stress element is introduced and two different types of a finite piecewise linear approximation of the dual elasticity problem are investigated on a polyhedral domain. Fot both types a priori error estimates in -norm and in -norm are established, provided the solution is smooth enough. These estimates are based on the fact that for any polyhedron there exists a strongly regular family of decomprositions into tetrahedra, which is proved in the paper, too.