In this thesis we investigate nonsmooth classical and continuum mechanics and its discretizations by means of variational numerical and geometric methods. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Next, we develop the f...

A variational multiscale method with multifractal subgrid-scale modeling is proposed for large-eddy simulation of turbulent flow. In the multifractal subgrid-scale modeling approach, the subgrid-scale velocity is evaluated from a multifractal description of the subgrid-scale vorticity, which is based on the multifractal scale similarity of gradient fields in turbulent flow. The multifractal sub...

We use the energetic formulation for rate–independent inelasticity which is based in the dissipation distance on the internal state. From this the incremental problems inherit a variational form which can be used to derive suitable relaxations. As a result we obtain a similar energetic formulation on the set of Young measures which then describes the evolution of microstructure.

This paper presents a mixed variational formulation and its discretization by finite elements of higher-order for the Signorini problem with Tresca friction. To guarantee the unique existence of the solution to the discrete mixed problem, a discrete inf-sup condition is proved. Moreover, a solution scheme based on the dual formulation of the problem is proposed. Numerical results confirm the th...

A numerical code for modeling crack propagation using the Cell Method is proposed. The Leon failure surface is used to compute the direction of crack propagation, and the new crack geometry is realized by an intra-element propagation technique. Automatic remeshing is then activated. Applications in Mode I, Mode II and Mixed Mode are presented to illustrate the robustness of the implementation. ...

We consider the tensorial diffusion equation, and address the discrete maximumminimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Ravi...

Data clustering, including problems such as finding network communities, can be put into a systematic framework by means of a Bayesian approach. Here we address the Bayesian formulation of the problem of finding hypergraph communities. We start by introducing a hypergraph generative model with a built-in group structure. Using a variational calculation we derive a variational Bayes algorithm, a...

We consider a logistic regression model with a Gaussian prior distribution over the parameters. We show that an accurate variational transformation can be used to obtain a closed form approximation to the posterior distribution of the parameters thereby yielding an approximate posterior predictive model. This approach is readily extended to binary graphical model with complete observations. For...

Selfdual variational theory – developed in [7] and [8] – allows for the superposition of appropriate boundary Lagrangians with “interior” Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in ...

We consider a logistic regression model with a Gaussian prior distribution over the parameters. We show that accurate variational techniques can be used to obtain a closed form posterior distribution over the parameters given the data thereby yielding a posterior predictive model. The results are readily extended to binary belief networks. For belief networks we also derive closed form posterio...