• Multiscale Problems
• Winter School

• Schedule
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• Poster

Multiscale processes pose severe challenges to scientific computation due to their very large number of degrees of freedom. We will focus on principles behind a variety of numerical multiscale methods, which have been developed to overcome this difficulty. As a background we will first briefly discuss analytical techniques. Averaging, geometrical optics and homogenization are of this type. In some numerical algorithms such as Multigrid and multipole all degrees of freedom are included.

Special multiscale properties of the approximated operators are exploited in order to reduce the computational complexity in the simulations. In other types of algorithms all or most of the degrees of freedom are originally included in the methods. These degrees of freedom are then compressed in order to generate a more computationally efficient algorithm. The multiscale finite element method is of this type. Finally we will introduce numerical strategies in which the finest scales are only well represented in small sub domains. In the heterogeneous multiscale method these sub domains supply information to a coarser discretization covering the full computational domain. The so-called equation free technique and super-parameterization are other similar methods.

Slides: Introduction to Computational Multiscale Modeling

The dynamic fracture of brittle solids is a particularly interesting collective interaction connecting both large and small length scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen.

With these issues in mind we discuss a new class of multi-scale models for solving problems of free crack propagation described by the peridynamic formulation. The peridynamic formulation is similar to molecular mechanics in that material points interact through short-range forces inside a prescribed horizon. The formulation allows for both continuous and discontinuous deformations associated with cracks. To illustrate the fundamentals we focus on a model for which the short-range forces between material points become unstable and soften beyond a critical relative displacement. This model is used to represent the dynamics at mesoscopic length scales. Analysis shows that the resulting mesoscopic dynamics is well posed.

It is possible to upscale this mesoscopic model and identify the relevant macroscopic dynamics across coarser length scales. Analysis shows that the associated macroscopic evolution has bounded energy given by the bulk and surface energies of classic brittle fracture mechanics. The macroscopic free crack evolution corresponds to the simultaneous evolution of the fracture surface and linear elastic displacement away from the crack set. The elastic moduli, wave speed, and energy release rate for the macroscopic evolution are explicitly determined by moments of the nonlocal potential energy. In this way we can make a connection between nonlocal short-range forces acting over small length scales and dynamic free crack evolution inside a brittle medium as observed at the macroscopic scale.

In lecture 1 we will address the physical and mathematical underpinnings for nonlocal fracture modeling.

In lecture 2 we will focus on mathematics of upscaling for nonlocal methods, including appropriate notions of weak convergence, compactness, and Gamma-convergence.

In lecture 3 we will focus on numerical simulation and numerical analysis of nonlocal methods.

Recorded Talks: Lecture 1 | Lecture 2 | Lecture 3

Many physical processes in micro-heterogeneous media such as modern composite and functional materials are described by partial differential equations (PDEs) with diffusion coefficients that represent complex material microstructures. Given the complexity of these processes, the key to reliably simulate some relevant classes of such processes involves the construction of appropriate computable macroscopic (homogenized or effective) models.

Numerical homogenization is a multiscale method for the derivation and simulation of such macroscopic models. This series of lectures reviews some two state-of-the-art techniques for numerical homogenization for a simple model problem. One technique is based on the efficient discretization of limiting models derived via the mathematical theory of homogenization. This requires structural assumptions on the coefficient such as periodicity and scale separation. The alternative approach is reliable for arbitrarily unstructured and rough coefficients.

Lecture 1 illustrates the model problem of oscillatory diffusion, the challenges for its numerical simulation, as well as the two numerical homogenization strategies.

Lecture 2 is devoted to a brief survey of constructive approaches in the mathematical theory of homogenization, which are valid under the structural assumption of local periodicity. By discretizing the resulting effective PDE with a diffusion coefficient given only implicitly, we will derive an efficient numerical homogenization method. We will study the error committed by this approach.

Lecture 3 presents an alternative numerical approach to homogenization that is solely based on the availability of operator-dependent subspaces with a quasi-local basis and approximation properties independent of oscillations and roughness of the diffusion coefficient. Analytical and experimental results will demonstrate the added value of this approach when compared to the previous one, i.e., its applicability, reliability, and accuracy in the absence of strong (unrealistic) assumptions such as periodicity and scale separation.

Notes: part1 | part2 | part3

After an introductory part explaining the potential of adaptive methods, we discuss a posteriori error estimation for finite element discretizations of elliptic PDEs, newest vertex bisection, and present the adaptive finite element loop.

Recorded Talk

Details of the proof of convergence of AFEM applied to elliptic PDEs will be presented. We introduce approximation classes, and prove that AFEMs converge with the best possible rate.

Recorded Talk

We show how bases for Sobolev spaces on general domains can be built from wavelets. These bases can be used for the optimal adaptive solution of well-posed linear and nonlinear operator equations. We discuss various applications including those to time-dependent PDEs. If time permits, then some details will be given about tensor product approximation and the reformulation of 2nd order PDEs as 1st order systems.

Recorded Talk