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Differential Equations, Continuum Mechanics and Numerical Methods
An Interdisciplinary Learning Seminar



Dr. Luca Heltai

Distributional Body Force Densities in Finite Element Approximations of Continuum Mechanics Problems

A number of problems in continuum mechanics are known for their singular behavior and for the difficulties connected to their numerical simulation. Some of them can be reformulated by introducing a distributional body force density that induces the desired singularities on the solution.
We present this idea in two different frameworks: fluid structure interaction and crack formation in linearly elastic materials. The natural numerical framework for the simulation of these distributional problems is the Finite Element Method. A number of examples that show the potentiality of this technique are presented and compared with existing methodologies.

Jonathan Pitt

Finite Element Implementation of a Linear Thermoelastic Material with Damage

Life prediction of structures and components is a fundamental responsibility of the engineering discipline. The field of continuum damage mechanics (CDM) answers this call by coupling the state of damage of a deformable body with the equations describing the body's motion. Thus, we have derived a dynamic coupled system, for a linear thermoelastic material, that allows the state of damage of the body to influence, as well as evolve according to, the motion of the body.
The resulting system is discretized using both Finite Differences (for time) and Finite Elements (for space). Mesh adaptivity has been implemented, as well as parallelization to accommodate the large number of degrees of freedoms. Simulation results in two and three dimensions will be presented, as well as the current status of the work.

Hengguang Li

Convergence rates of the multigrid method on graded meshes for corner-like singularities

It is well known that the non-smoothness of the boundary of a 2-D domain and changes of boundary conditions of an elliptic equation may lead to singular solutions in a certain Sobolev space. Graded or adaptive meshes are widely used to recover the optimal convergence rates of the numerical solutions from the finite element method, while the multigrid method has been proved to converge uniformly on uniform meshes. Therefore, a nature question is raised: does the multigrid method still converge uniformly on graded meshes of good quality? We here shall analyze the convergence rates of the multigrid method and give a positive answer to the question. The theoretical framework is set up by using the weighted Sobolev space and the method of subspace corrections. A special elliptic projection decomposition estimate on weighted Sobolev spaces is implemented. Our goal here is to show that the multigrid V-cycle converges uniformly for piecewise linear functions with standard smoothers (Richardson, weighted Jacobi, Gauss-Seidel, etc.) on each level of subspaces. In addition, we have a similar argument on high-order polynomials.

Prof. Francesco Costanzo

A Discontinuous Galerkin Space-Time Formulation for Linear Elastodynamics with Moving Surfaces of Strain Discontinuity

A discontinuous Galerkin Space-Time FEM formulation is presented for the solution of elasto-dynamics problems in the presence of moving surfaces of stress/strain discontinuity such as those encountered in dynamic solid/solid phase transitions. The formulation is time-discontinuous in the sense that discontinuities in the primary unknown field are allowed across a discrete set of time increments. Special focus is devoted to the discussion of the unconditional stability of the formulation. A generalization of the formulation to linear thermo-elasto-dynamics is also presented along with various numerical results to demonstrate the method's capabilities and a "physically-based" method for h-adaptive refinement of the space-time grid.

Prof. Daniele Boffi

Approximation of variationally posed eigenvalue problems

In the first part of the talk, we review the use of Galerkin method for the approximation of the eigenpairs of an elliptic operator. Then, we discuss the extensions to more general situations and, in particular, the approximation theory of eigenproblems in mixed form is described in more detail.
Starting from the curl curl Maxwell's operator, we use as a template the mixed approximation of Laplace eigenpairs. Necessary and sufficient conditions for the convergence of eigenvalues/eigenfunctions arising from mixed approximation of partial differential equations are presented.

The seminar is sponsored by:
Center for Computational Mathematics and Applications
Laboratory for Parallel Computational Mechanics