On the convergence of the rescaled localized radial basis function method

The rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture

Radial basis function interpolation for black-box multi-physics simulations

Interpolation based on radial basis functions (RBF) is a standard data map- ping method used in multi-physics coupling. It works on scattered data without requiring additional mesh topology or neighborhood information of support points. However, sys- tem matrices of the equations for the coefficients tend to be ill-conditioned. In this work, we illustrate the problem by a simple example and discuss possible remedies. Furthermore, we investigate the numerical performance of this method on uniform and non-uniform meshes with a particular focus on the coupling of black-box components where typically no information about the underlying discretization can be extracted. Radial basis func- tion interpolation usually uses an enhancement of the radial basis functions by a global polynomial in order to properly capture constant components and linear trends in the given data. We present a method that determines this polynomial independent from the radial basis function ansatz, which substantially improves the condition number of the remaining RBF system. Furthermore, we show that a rescaling approach can be used to either increase the accuracy or improve the condition number even further by choosing radial basis functions with a smaller support radius. The results represent an intermediate state with the aim to be integrated into the multi-physics coupling library preCICE

Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: application to cardiac electromechanics

The accurate robust and efficient transfer of the deformation gradient tensor between meshes of different resolution is crucial in cardiac electromechanics simulations. We present a novel method that combines rescaled localized Radial Basis Function (RBF) interpolation with Singular Value Decomposition (SVD) to preserve the positivity of the determinant of the deformation gradient tensor. The method involves decomposing the evaluations of the tensor at the quadrature nodes of the source mesh into rotation matrices and diagonal matrices of singular values; computing the RBF interpolation of the quaternion representation of rotation matrices and the singular value logarithms; reassembling the deformation gradient tensors at quadrature nodes of the destination mesh, to be used in the assembly of the electrophysiology model equations. The proposed method overcomes limitations of existing interpolation methods, including nested intergrid interpolation and RBF interpolation of the displacement field, that may lead to the loss of physical meaningfulness of the mathematical formulation and then to solver failures at the algebraic level, due to negative determinant values. The proposed method enables the transfer of solution variables between finite element spaces of different degrees and shapes and without stringent conformity requirements between different meshes, enhancing the flexibility and accuracy of electromechanical simulations. Numerical results confirm that the proposed method enables the transfer of the deformation gradient tensor, allowing to successfully run simulations in cases where existing methods fail. This work provides an efficient and robust method for the intergrid transfer of the deformation gradient tensor, enabling independent tailoring of mesh discretizations to the particular characteristics of the physical components concurring to the of the multiphysics model.Comment: 24 pages; 11 figure

Efficient computation of partition of unity interpolants through a block-based searching technique

In this paper we propose a new efficient interpolation tool, extremely suitable for large scattered data sets. The partition of unity method is used and performed by blending Radial Basis Functions (RBFs) as local approximants and using locally supported weight functions. In particular we present a new space-partitioning data structure based on a partition of the underlying generic domain in blocks. This approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points, leading to an optimized searching routine. Complexity analysis and numerical experiments in two- and three-dimensional interpolation support our findings. Some applications to geometric modelling are also considered. Moreover, the associated software package written in \textsc{Matlab} is here discussed and made available to the scientific community

The INTERNODES method for the treatment of non-conforming multipatch geometries in Isogeometric Analysis

In this paper we apply the INTERNODES method to solve second order elliptic problems discretized by Isogeometric Analysis methods on non-conforming multiple patches in 2D and 3D geometries. INTERNODES is an interpolation-based method that, on each interface of the configuration, exploits two independent interpolation operators to enforce the continuity of the traces and of the normal derivatives. INTERNODES supports non-conformity on NURBS spaces as well as on geometries. We specify how to set up the interpolation matrices on non-conforming interfaces, how to enforce the continuity of the normal derivatives and we give special attention to implementation aspects. The numerical results show that INTERNODES exhibits optimal convergence rate with respect to the mesh size of the NURBS spaces an that it is robust with respect to jumping coefficients.Comment: Accepted for publication in Computer Methods in Applied Mechanics and Engineerin

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

Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces

One of the major challenges of coupled problems is to manage nonconforming meshes at the interface between two models and/or domains, due to different numerical schemes or domain discretizations employed. Moreover, very often complex submodels depend on (e.g., physical or geometrical) parameters, thus making the repeated solutions of the coupled problem through high-fidelity, full-order models extremely expensive, if not unaffordable. In this paper, we propose a reduced order modeling (ROM) strategy to tackle parametrized one-way coupled problems made by a first, master model and a second, slave model; this latter depends on the former through Dirichlet interface conditions. We combine a reduced basis method, applied to each subproblem, with the discrete empirical interpolation method to efficiently interpolate or project Dirichlet data across either conforming or non-conforming meshes at the domains interface, building a low-dimensional representation of the overall coupled problem. The proposed technique is numerically verified by considering a series of test cases involving both steady and unsteady problems, after deriving a posteriori error estimates on the solution of the coupled problem in both cases. This work arises from the need to solve staggered cardiac electrophysiological models and represents the first step towards the setting of ROM techniques for the more general two-way Dirichlet-Neumann coupled problems solved with domain decomposition sub-structuring methods, when interface non-conformity is involved

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