A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles - rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities.
Partition of unity methods (PUMs) on graphs are simple and highly adaptive auxiliary tools for graph signal processing. Based on a greedy-type metric clustering and augmentation scheme, we show how a partition of unity can be generated in an efficient way on graphs. We investigate how PUMs can be combined with a local graph basis function (GBF) approximation method in order to obtain low-cost global interpolation or classification schemes. From a theoretical point of view, we study necessary prerequisites for the partition of unity such that global error estimates of the PUM follow from corres…
The adaptive partition of unity interpolation method, introduced by Aiton and Driscoll, using Chebyshev local interpolants, is explored for interpolating functions with sharp gradients representing two-medium problems. For functions that evolve under vector fields, the partition of unity patches (covers) can be shifted and resized to follow the changing dynamics of local profiles. The method is tested for selected 1D and 2D two-medium problems with linear divergence-free vector fields. In those cases, the volume fraction in each patch contributing to volume conservation throughout the domain c…
In this work, a space-time scheme for goal-oriented a posteriori error estimation is proposed. The error estimator is evaluated using a partition-of-unity dual-weighted residual method. As application, a low mach number combustion equation is considered. In some numerical tests, different interpolation variants are investigated, while observing convergence orders and effectivity indices between true errors (obtained on a sufficiently refined mesh) and the error estimator.
Partition of Unity Extension of Functions on Complex Domains
KTH Mathematics, Swedish e–Science Research Centre（KTH数学系，瑞典e-科学研究中心）
本文提出了一种高效的Partition of Unity扩展（PUX）算法，用于在二维复多连通域上构造具有所需正则性的函数扩展。该方法通过在边界上放置重叠分区，并利用光滑径向基函数在每个区域计算局部扩展，从而实现全局扩展。通过构造合适的Partition of Unity函数，可以控制扩展函数的正则性，并在求解多连通域上的泊松方程时验证了该方法的性能，误差收敛至10^-14。
In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method is benefited from a direct discretization approach and is called the `direct RBF partition of unity (D-RBF-PU)' method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and…
Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems
Approximation theorists have established best-in-class optimal approximation rates of deep neural networks by utilizing their ability to simultaneously emulate partitions of unity and monomials. Motivated by this, we propose partition of unity networks (POUnets) which incorporate these elements directly into the architecture. Classification architectures of the type used to learn probability measures are used to build a meshfree partition of space, while polynomial spaces with learnable coefficients are associated to each partition. The resulting hp-element-like approximation allows use of a f…
In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier-Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor-Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition of unity based error localization. We derive th…
Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set) approximation for an oriented point cloud using only information about (approximate) normals to the surface. The technique exploits the fundamental result from vector calculus that the normals to an implicit surface are curl-free. By using a curl-free radial basis function (RBF) interpolation of the normals, we can extract a potential for the vector field whose ze…
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized {nonlinear} elliptic partial differential equations (PDEs). CB-pMOR is designed to deal with large-scale problems for which full-order solves are not affordable in a reasonable time frame or parameters' variations induce topology changes that prevent the application of monolithic pMOR techniques. We rely on the partition-of-unity method (PUM) to devise global approximation spaces from local reduced spaces, and on Galerkin projection to compute the global state estimate. We propose a randomiz…
In this work, we implement goal-oriented error control and spatial mesh adaptivity for stationary fluid-structure interaction. The a posteriori error estimator is realized using the dual-weighted residual method in which the adjoint equation arises. The fluid-structure interaction problem is formulated within a variational-monolithic framework using arbitrary Lagrangian-Eulerian coordinates. The overall problem is nonlinear and solved with Newton's method. We specifically consider the FSI-1 benchmark problem in which quantities of interest include the elastic beam displacements, drag, and lift…
Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfr…
A parametric study on the buckling of functionally graded material plates with internal discontinuities using the partition of unity method
The University of New South Wales（新南威尔士大学）；Indian Institute of Science（印度科学学院）；Mahindra Satyam Computer Services Ltd.（马恒达萨扬计算机服务有限公司）；Dr. Mahalingam College of Engineering and Technology（德拉·马哈林甘工程技术学院）
Anisotropic weights for RBF-PU interpolation with subdomains of variable shapes
Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova（帕多瓦大学）；Department of Mathematics “Giuseppe Peano”, University of Torino（都灵大学）；Department of Applied Mathematics and Statistics, Colorado School of Mines（科罗拉多矿业学院）；SigOpt, Inc.（SigOpt公司）
We focus on the finite element method computations with higher-order C1 continuity basis functions that preserve the partition of unity. We show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is a Petrov-Galerkin formulation with piece…
We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed $C^0$ continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in $\mathbb R^3$ is used to define a set of overlapping charts…
