Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

Meshless boundary particle methods for boundary integral equations and meshfree particle methods for plates

For approximating the solution of partial differential equations (PDE), meshless methods have been introduced to alleviate the difficulties arising in mesh generation using the conventional Finite Element Method (FEM). Many meshless methods intro- duced lack the Kronecker delta property making them inefficient in handling essential boundary conditions. Oh et al. developed several meshfree shape functions that have the Kronecker delta property. Boundary Element Methods (BEM) solve a boundary integral equation (BIE) which is equivalent to the PDE, thus reducing the dimen- sionality of the problem by one and the amount of computation when compared to FEM. In this dissertation, three meshless collocation based boundary element meth- ods are introduced: meshfree reproducing polynomial boundary particle method (RPBPM), patch-wise RPBPM, and patch-wise reproducing singularity particle method (RSBPM). They are applied to the Laplace equation for convex and non-convex do- mains in two and three dimensions for problems with and without domain singulari- ties. Electromagnetic wave propagation through photonic crystals is governed by Maxwell’s equations in the frequency domain. Under certain conditions, it can be shown that the wave propagation is also governed by Helmholtz equation. Patch-wise RPBPM is applied to the two dimensional Helmholtz equation and used to model electromagnetic wave propagation though lattices of photonic crystals. For thin plate problems, using the Kirchoff hypothesis, the three dimensional elasticity equations are reduced to a fourth order PDE for the vertical displacement. Conventional FEM has difficulties in solving this because the basis functions are required to have continuous partial derivatives. Suggestions are to use Hermite based elements which are difficult to implement. Using a partition of unity, some special shape functions are developed for thin plates with simple support or clamped bound- ary conditions. This meshless method for thin plates is then tested and the results are reported

A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations

A GENERALIZED FINITE DIFFERENCE METHOD FOR TRANSIENT HEAT CONDUCTION ANALYSIS-SHORT COMMUNICATION

This short communication presents a meshless local B-spline basis functions-finite difference (FD) method for transient heat conduction analysis. The method is truly meshless as only scattered nodal distribution is required in the problem domain. It is also simple and efficient to program. As it has the Kronecker delta property, the imposition of boundary conditions can be incorporated efficiently. In the method, any governing equations are discretized by B-spline approximation in the spirit of FD technique using local B-spline collocation. It hence belongs to a generalized FD method, in which any derivative at a point or node is stated as neighbouring nodal values based on the B-spline interpolants. Numerical results show the effectiveness and efficiency of the meshless method for analysis of transient heat conduction in complex domain

Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems

This is the accepted version of the following article: [Canales, D., Leygue, A., Chinesta, F., González, D., Cueto, E., Feulvarch, E., Bergheau, J. -M., and Huerta, A. (2016) Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems. Int. J. Numer. Meth. Engng, 108: 971–989. doi: 10.1002/nme.5240.], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5240/fullThis paper proposes a generalized finite element method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional finite element methods at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing proper generalized decompositions. Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics that depend on many parameters, such as welding. After introducing the vademecum-generalized finite element method formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes.Peer ReviewedPostprint (author's final draft

Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method

Set-free Markov state model building

Molecular dynamics (MD) simulations face challenging problems since the time scales of interest often are much longer than what is possible to simulate; and even if sufficiently long simulations are possible the complex nature of the resulting simulation data makes interpretation difficult. Markov State Models (MSMs) help to overcome these problems by making experimentally relevant time scales accessible via coarse grained representations that also allow for convenient interpretation. However, standard set-based MSMs exhibit some caveats limiting their approximation quality and statistical significance. One of the main caveats results from the fact that typical MD trajectories repeatedly re-cross the boundary between the sets used to build the MSM which causes statistical bias in estimating the transition probabilities between these sets. In this article, we present a set-free approach to MSM building utilizing smooth overlapping ansatz functions instead of sets and an adaptive refinement approach. This kind of meshless discretization helps to overcome the recrossing problem and yields an adaptive refinement procedure that allows us to improve the quality of the model while exploring state space and inserting new ansatz functions into the MSM