Hexahedral-dominant meshing

This article introduces a method that generates a hexahedral-dominant mesh from an input tetrahedral mesh.It follows a three-steps pipeline similar to the one proposed by Carrier-Baudoin et al.:(1) generate a frame field; (2) generate a pointset P that is mostly organized on a regulargrid locally aligned with the frame field; and (3) generate thehexahedral-dominant mesh by recombining the tetrahedra obtained from the constrained Delaunay triangulation of P.For step (1), we use a state of the art algorithm to generate a smooth frame field. For step (2), weintroduce an extension of Periodic Global Parameterization to the volumetric case. As compared withother global parameterization methods (such as CubeCover), our method relaxes some global constraintsand avoids creating degenerate elements, at the expense of introducing some singularities that aremeshed using non-hexahedral elements. For step (3), we build on the formalism introduced byMeshkat and Talmor, fill-in a gap in their proof and provide a complete enumeration of all thepossible recombinations, as well as an algorithm that efficiently detects all the matches in a tetrahedral mesh.The method is evaluated and compared with the state of the art on adatabase of examples with various mesh complexities, varying fromacademic examples to real industrial cases. Compared with the methodof Carrier-Baudoin et al., the method results in better scoresfor classical quality criteria of hexahedral-dominant meshes(hexahedral proportion, scaled Jacobian, etc.). The methodalso shows better robustness than CubeCover and its derivativeswhen applied to complicated industrial models

The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques

The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems

AMM: Adaptive Multilinear Meshes

We present Adaptive Multilinear Meshes (AMM), a new framework that significantly reduces the memory footprint compared to existing data structures. AMM uses a hierarchy of cuboidal cells to create continuous, piecewise multilinear representation of uniformly sampled data. Furthermore, AMM can selectively relax or enforce constraints on conformity, continuity, and coverage, creating a highly adaptive and flexible representation to support a wide range of use cases. AMM supports incremental updates in both spatial resolution and numerical precision establishing the first practical data structure that can seamlessly explore the tradeoff between resolution and precision. We use tensor products of linear B-spline wavelets to create an adaptive representation and illustrate the advantages of our framework. AMM provides a simple interface for evaluating the function defined on the adaptive mesh, efficiently traversing the mesh, and manipulating the mesh, including incremental, partial updates. Our framework is easy to adopt for standard visualization and analysis tasks. As an example, we provide a VTK interface, through efficient on-demand conversion, which can be used directly by corresponding tools, such as VisIt, disseminating the advantages of faster processing and a smaller memory footprint to a wider audience. We demonstrate the advantages of our approach for simplifying scalar-valued data for commonly used visualization and analysis tasks using incremental construction, according to mixed resolution and precision data streams

[Activity of Institute for Computer Applications in Science and Engineering]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science

Innovative Approaches to the Numerical Approximation of PDEs

This workshop was about the numerical solution of PDEs for which classical approaches, such as the finite element method, are not well suited or need further (theoretical) underpinnings. A prominent example of PDEs for which classical methods are not well suited are PDEs posed in high space dimensions. New results on low rank tensor approximation for those problems were presented. Other presentations dealt with regularity of PDEs, the numerical solution of PDEs on surfaces, PDEs of fractional order, numerical solvers for PDEs that converge with exponential rates, and the application of deep neural networks for solving PDEs

Parallel Algorithms for the Solution of Large-Scale Fluid-Structure Interaction Problems in Hemodynamics

This thesis addresses the development and implementation of efficient and parallel algorithms for the numerical simulation of Fluid-Structure Interaction (FSI) problems in hemodynamics. Indeed, hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the arterial wall. The simulation of fluid-structure interaction problems requires the approximation of a coupled system of Partial Differential Equations (PDEs) and the set up of efficient numerical solution strategies. Blood is modeled as an incompressible Newtonian fluid whose dynamics is governed by the Navier-Stokes equations. Different constituive models are used to describe the mechanical response of the arterial wall; specifically, we rely on hyperelastic isotropic and anistotropic material laws. The finite element method is used for the space discretization of both the fluid and structure problems. In particular, for the Navier-Stokes equations we consider a semi-discrete formulation based on the Variational Multiscale (VMS) method. Among a wide range of possible solution strategies for the FSI problem, here we focus on strongly coupled monolithic approaches wherein the nonlinearities are treated in a fully implicit mode. To cope with the high computational complexity of the three dimensional FSI problem, a parallel solution framework is often mandatory. To this end, we develop a new block parallel preconditioner for the coupled linearized FSI system obtained after space and time discretization. The proposed preconditioner, named FaCSI, exploits the factorized form of the FSI Jacobian matrix, the use of static condensation to formally eliminate the interface degrees of freedom of the fluid equations, and the use of a SIMPLE preconditioner for unsteady Navier-Stokes equations. In FSI problems, the different resolution requirements in the fluid and structure physical domains, as well as the presence of complex interface geometries make the use of matching fluid and structure meshes problematic. In such situations, it is much simpler to deal with discretizations that are nonconforming at the interface, provided however that the matching conditions at the interface are properly fulfilled. In this thesis we develop a novel interpolation-based method, named INTERNODES, for numerically solving partial differential equations by Galerkin methods on computational domains that are split into two (or several) subdomains featuring nonconforming interfaces. By this we mean that either a priori independent grids and/or local polynomial degrees are used to discretize each subdomain. INTERNODES can be regarded as an alternative to the mortar element method: it combines the accuracy of the latter with the easiness of implementation in a numerical code. The aforementioned techniques have been applied for the numerical simulation of large-scale fluid-structure interaction problems in the context of biomechanics. The parallel algorithms developed showed scalability up to thousands of cores utilized on high performance computing machines

Extended Variational Formulation for Heterogeneous Partial Differential Equations

We address the coupling of an advection equation with a diffusion-advection equation, for solutions featuring boundary layers. We consider non-overlapping domain decompositions and we face up the heterogeneous problem using an extended variational formulation. We will prove the equivalence between the latter formulation and a treatment based on a singular perturbation theory. An exhaustive comparison in terms of solution and computational efficiency between these formulations is carried ou

Big-Data Science in Porous Materials: Materials Genomics and Machine Learning

By combining metal nodes with organic linkers we can potentially synthesize millions of possible metal organic frameworks (MOFs). At present, we have libraries of over ten thousand synthesized materials and millions of in-silico predicted materials. The fact that we have so many materials opens many exciting avenues to tailor make a material that is optimal for a given application. However, from an experimental and computational point of view we simply have too many materials to screen using brute-force techniques. In this review, we show that having so many materials allows us to use big-data methods as a powerful technique to study these materials and to discover complex correlations. The first part of the review gives an introduction to the principles of big-data science. We emphasize the importance of data collection, methods to augment small data sets, how to select appropriate training sets. An important part of this review are the different approaches that are used to represent these materials in feature space. The review also includes a general overview of the different ML techniques, but as most applications in porous materials use supervised ML our review is focused on the different approaches for supervised ML. In particular, we review the different method to optimize the ML process and how to quantify the performance of the different methods. In the second part, we review how the different approaches of ML have been applied to porous materials. In particular, we discuss applications in the field of gas storage and separation, the stability of these materials, their electronic properties, and their synthesis. The range of topics illustrates the large variety of topics that can be studied with big-data science. Given the increasing interest of the scientific community in ML, we expect this list to rapidly expand in the coming years.Comment: Editorial changes (typos fixed, minor adjustments to figures