Strong-form meshless methods received much attention in recent years and are being extensively researched and applied to a wide range of problems in science and engineering. However, the solution of ...elasto-plastic problems has proven to be elusive because of often non-smooth constitutive relations between stress and strain. The novelty in tackling them is the introduction of virtual finite difference stencils to formulate a hybrid radial basis function generated finite difference (RBF-FD) method, which is used to solve small-strain von Mises elasto-plasticity for the first time by this original approach. The paper further contrasts the new method to two alternative legacy RBF-FD approaches, which fail when applied to this class of problems. The three approaches differ in the discretization of the divergence operator found in the balance equation that acts on the non-smooth stress field. Additionally, an innovative stabilization technique is employed to stabilize boundary conditions and is shown to be essential for any of the approaches to converge successfully. Approaches are assessed on elastic and elasto-plastic benchmarks where admissible ranges of newly introduced free parameters are studied regarding stability, accuracy, and convergence rate.
•Three meshless strong form numerical method variants are employed to address the small strain von-Mises elasto-plasticity.•A new approach combining virtual FD stencils with RBF-FD is proposed and characterized on a set of 2D benchmarks.•We found that the new approach is superior compared to the other two for accurately solving elasto-plastic problems.
This work extends our research on the strong-form meshless Radial Basis Function - Finite Difference (RBF-FD) method for solving non-linear visco-plastic mechanical problems. The polyharmonic splines ...with second-order polynomial augmentation are used for the shape functions. Their coefficients are determined by collocation. Three different approaches (direct, composed, and hybrid) are used for the numerical evaluation of the divergence operator in the equilibrium equation. They are presented and assessed for a visco-plastic material model with continuously differentiable material properties. It is shown that the direct approach is not suitable in this respect. In comparison to the previously investigated elasto-plasticity, it is shown that the composed approach can successfully cope with visco-plastic problems and is found to be even more accurate than the hybrid approach, which has previously proven to be most stable and effective in solving elasto-plasticity. This work extends the applicability of strong-form RBF-FD methods and opens up new areas of modelling non-linear solid mechanics.
•Three meshless strong form numerical method variants employed on visco-plasticity.•RBF-FD approaches are characterised on two different 2D visco-plastic benchmarks.•Two approaches work well and produce results comparable to FEM.
•Easy approximation for derivatives on a sphere with RBF-FD splines+2D polynomials.•Very short and simple codes, even when implemented to high orders of accuracy.•Highly effective method, in terms of ...accuracy, convergence, and computational time.•Tested on standard benchmarks for tracer transport in 2D/3D spherical geometries.•Shown to be very competitive against other methods as FV, DG, SE, and RBF-PUM.
This work presents a numerical algorithm for using radial basis function-generated finite differences (RBF-FD) to solve partial differential equations (PDEs) on S2 using polyharmonic splines with added polynomials defined in a 2D plane (PHS+Poly). We introduce a novel method for calculating RBF-FD PHS+Poly differentiation weights on S2 using first a Householder reflection and then a projection onto the tangent plane. The new PHS+Poly RBF-FD method is implemented on two standard test cases: 1) solid body rotation on S2 and 2) 3D tracer transport within Earth's atmosphere. Compared to existing methods (including those in the RBF literature) at similar resolutions, this approach requires fewer degrees of freedom and is algorithmically much simpler. A MATLAB code to implement the method is included in the Appendix.
In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not ...requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher accuracy. Still, the discretization matrix is fairly sparse. As a model problem, we consider the pricing of European options in 2D. Since such options have a discontinuity in the first derivative of the payoff function which prohibits high order convergence, we smooth this function using an established technique for Cartesian grids. Numerical experiments show that we acquire a fourth order scheme in space, both for the uniform and the nonuniform node layouts that we use. The high order method with the nonuniform node layout achieves very high accuracy with relatively few nodes. This renders the potential for solving pricing problems in higher spatial dimensions since the computational memory and time demand become much smaller with this method compared to standard techniques.
To overcome a significant challenge in traditional parameterized level set methods based on globally supported radial basis functions, we propose employing a local differentiation construction of ...radial basis functions using finite difference, a technique previously applied to solving partial differential equations but novel in the context of topology optimization. We present a novel parameterized level set method for structural topology optimization of compliance minimization and compliant mechanism, with the main aim of reducing computational costs associated with fully dense matrices when approximating systems with a large number of collocation points. The new scheme implemented with rectangular mesh elements and polygonal mesh generation accommodates both rectangular and complex design domains. Numerical results are provided to demonstrate the algorithm's effectiveness.
•Local meshless radial basis functions are used in level set method for topology optimization.•Gaussian and multiquadric radial basis functions are combined with finite differences.•Compliance minimization and compliant mechanism are solved numerically.•Polygonal meshes are used for complex design domains.•Numerical results show the new parameterized level set method is effective and efficient.
The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial ...differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.
Summary
In this work, we are concerned with radial basis function–generated finite difference (RBF‐FD) approximations. Numerical error estimates are presented for stabilized flat Gaussians ...(RBF(SGA)‐FD) and polyharmonic splines with supplementary polynomials (RBF(PHS)‐FD) using some analytical solutions of the Poisson equation in a square domain. Both structured and unstructured point clouds are employed for evaluating the influence of cloud refinement, size of local supports, and maximal permissible degree of the polynomials in RBF(PHS)‐FD. High order of accuracy was attained with both RBF(SGA)‐FD and RBF(PHS)‐FD especially for unstructured clouds. Absolute errors in the first and second derivatives were also estimated at all points of the domain using one of the analytical solutions. For RBF(SGA)‐FD, this test showed the occurrence of improprieties of some decentered supports localized on boundary neighborhoods. This phenomenon was not observed with RBF(PHS)‐FD.
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