Showing papers on "Meshfree methods published in 2015"

CutFEM: Discretizing geometry and partial differential equations

Abstract: We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

A Radial Basis Function Partition of Unity Collocation Method for Convection---Diffusion Equations Arising in Financial Applications

Abstract: Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF---PUM) proposed here. The objective of this paper is to establish that RBF---PUM is viable for parabolic PDEs of convection---diffusion type. The stability and accuracy of RBF---PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection---diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF---PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF---PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

A Survey of Physically Based Simulation of Cuts in Deformable Bodies

Abstract: Virtual cutting of deformable bodies has been an important and active research topic in physically based modelling and simulation for more than a decade. A particular challenge in virtual cutting is the robust and efficient incorporation of cuts into an accurate computational model that is used for the simulation of the deformable body. This report presents a coherent summary of the state of the art in virtual cutting of deformable bodies, focusing on the distinct geometrical and topological representations of the deformable body, as well as the specific numerical discretizations of the governing equations of motion. In particular, we discuss virtual cutting based on tetrahedral, hexahedral and polyhedral meshes, in combination with standard, polyhedral, composite and extended finite element discretizations. A separate section is devoted to meshfree methods. Furthermore, we discuss cutting-related research problems such as collision detection and haptic rendering in the context of interactive cutting scenarios. The report is complemented with an application study to assess the performance of virtual cutting simulators.

The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods

Abstract: The present paper is devoted to the numerical solution of the Cahn–Hilliard (CH) equation in one, two and three-dimensions. We will apply two different meshless methods based on radial basis functions (RBFs). The first method is globally radial basis functions (GRBFs) and the second method is based on radial basis functions differential quadrature (RBFs-DQ) idea. In RBFs-DQ, the derivative value of function with respect to a point is directly approximated by a linear combination of all functional values in the global domain. The main aim of this method is the determination of weight coefficients. GRBFs replace the function approximation into the partial differential equation directly. Also, the coefficients matrix which arises from GRBFs is very ill-conditioned. The use of RBFs-DQ leads to the improvement of the ill-conditioning of interpolation matrix RBFs. The boundary conditions of the mentioned problem are Neumann. Thus, we use DQ method directly on the boundary conditions, which easily implements RBFs-DQ on the irregular points and regions. Here, we concentrate on Multiquadrics ( MQ ) as a radial function for approximating the solution of the mentioned equation. As we know this radial function depends on a constant parameter called shape parameter. The RBFs-DQ can be implemented in a parallel environment to reduce the computational time. Moreover, to obtain the error of two techniques with respect to the spatial domain, a predictor–corrector scheme will be applied. Finally, the numerical results show that the proposed methods are appropriate to solve the one, two and three-dimensional Cahn-Hilliard (CH) equations.

The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank–Nicolson scheme and the method of lines (MOL)

Abstract: MHD equations have many applications in physics and engineering. The model is coupled equations in velocity and magnetic field and has a parameter namely Hartmann. The value of Hartmann number plays an important role in the equations. When this parameter increases, using different meshless methods makes the oscillations in velocity near the boundary layers in the region of the problem. In the present paper a numerical meshless method based on radial basis functions (RBFs) is provided to solve MHD equations. For approximating the spatial variable, a new approach which is introduced by Bozzini et al. (2015) is applied. The method will be used here is based on the interpolation with variably scaled kernels. The methodology of the new technique is defining the scale function c on the domain Ω ⊂ R d . Then the interpolation problem from the data locations x j ∈ R d transforms to the new interpolation problem in the data locations ( x j , c ( x j ) ) ∈ R d + 1 (Bozzini et al., 2015). The radial kernels used in the current work are Multiquadrics (MQ), Inverse Quadric (IQ) and Wendland’s function. Of course the latter one is based on compactly supported functions. To discretize the time variable, two techniques are applied. One of them is the Crank–Nicolson scheme and another one is based on MOL. The numerical simulations have been carried out on the square and elliptical ducts and the obtained numerical results show the ability of the new method for solving this problem. Also in appendix, we provide a computational algorithm for implementing the new technique in MATLAB software.

Hybridization of the Wave Propagation Model SWASH and the Meshfree Particle Method SPH for Real Coastal Applications

Abstract: Two computational models, Simulating WAve till SHore (SWASH) and DualSPHysics, with different computational costs and capabilities have been hybridized in this work. SWASH is a time-domain wave mod...

Local radial basis function collocation method for linear thermoelasticity in two dimensions

Abstract: Purpose – The purpose of this paper is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM. Design/methodology/approach – Linear thermoelastic problems for homogenous isotropic body in two dimensions are considered. The stationary stress equilibrium equation is written in terms of deformation field. The domain and boundary can be discretized with arbitrary positioned nodes where the solution is sought. Each of the nodes has its influence domain, encompassing at least six neighboring nodes. The unknown displacement field is collocated on local influence domain nodes with shape functions that consist of a linear combination of multiquadric radial basis functions and monomials. The boundary conditions are analytically satisfied on the influence domains which co...

Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions

Abstract: In the present paper, two numerical methods are analyzed for the solution of two-dimensional Poisson equation with two different types of nonlocal boundary conditions. The first numerical method is a collocation method based on Haar wavelet whereas the second numerical method is a meshless method based on different types of radial basis functions (RBFs). A two-point boundary condition and an integral boundary condition are the two types of nonlocal boundary conditions considered in the present work. For the collocation method based on Haar wavelet a new approach is formulated which involves the approximation of a fourth order mixed derivative by a Haar expansion which is integrated subsequently to get wavelet approximation of the solution. For the meshless method based on RBFs, the algorithm is implemented using two different splitting schemes (with and without shape parameter splitting) for numerical solution of the model. The comparative analysis of the meshless methods with and without shape parameter splitting scheme is performed between themselves as well as with the Haar wavelet. Accuracy and efficiency wise performance is confirmed through application of the algorithms on the benchmark tests.

A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model

Abstract: In this paper local radial basis functions (LRBFs) collocation method is proposed for solving the (Patlak-) Keller–Segel model. We use the Crank–Nicolson difference scheme for the time derivative to obtain a finite difference scheme with respect to the time variable for the Keller–Segel model. Then we use the local radial basis functions (LRBFs) collocation method to approximate the spatial derivative. We obtain the numerical results for the mentioned model. As we know, recently some approaches presented for preventing the blow up of the cell density. In the current paper we use the multiquadric (MQ) radial basis function. The aim of this paper is to show that the meshless methods based on the local RBFs collocation approach are also suitable for solving models that have the blow up of the cell density. Also, six test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

A coupled smoothed finite element method (S-FEM) for structural-acoustic analysis of shells

Abstract: In this paper, a coupled smoothed finite element method (S-FEM) is developed to deal with the structural-acoustic problems consisting of a shell configuration interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements that can be generated automatically for any complicated geometries are adopted to discretize the problem domain. A gradient smoothing technique (GST) is introduced to perform the strain smoothing operation. The discretized system equations are obtained using the smoothed Galerkin weakform, and the numerical integration is applied over the further formed edge-based and face-based smoothing domains, respectively. To extend the edge-based smoothing operation from plate structure to shell structure, an edge coordinate system is defined local on the edges of the triangular element. Numerical examples of a cylinder cavity attached to a flexible shell and an automobile passenger compartment have been conducted to illustrate the effectiveness and accuracy of the coupled S-FEM for structural-acoustic problems.

A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation

Abstract: In the current work, numerical solution of a two-dimensional fractional evolution equation has been investigated by using two different aspects of strong form meshless methods. In the first method a time discretization approach and a numerical technique based on the convolution sum are employed to approximate the appearing time derivative and fractional integral operator, respectively. It has been proven analytically that the time discretization scheme is unconditionally stable. Then a meshfree collocation method based on the radial basis functions is used for solving resulting time-independent discretization problem. As the second approach, a fully Kansa׳s meshfree method based on the Gaussian radial basis function is formulated and well-used directly for solving the governing problem. In this technique an explicit formula to approximate the fractional integral operator is computed. The given techniques are used to solve two examples of problem. The computed approximate solutions are reported through the tables and figures, also these results are compared together and with the other available results. The presented results demonstrate the validity, efficiency and accuracy of the formulated techniques.

The Diffused Vortex Hydrodynamics Method

Abstract: A new Particle Vortex Method, called Diffused Vortex Hydrodynamics (DVH), is presented in this paper. The DVH is a meshless method characterized by the use of a regular distribution of points close to a solid surface to perform the vorticity diffusion process in the boundary layer regions. This redistribution avoids excessive clustering or rarefaction of the vortex particles providing robustness and high accuracy to the method. The generation of the regular distribution of points is performed through a packing algorithm which is embedded in the solver. The packing algorithm collocates points regularly around body of arbitrary shape allowing an exact enforcement on the solid surfaces of the no-slip boundary condition. The present method is tested and validated on different problems of increasing complexities up to flows with Reynolds number equal to 100,000 (without using any subgrid-scale turbulence model).

Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method

Abstract: The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.

An Edge-Based Smoothed Discrete Shear Gap Method Using the C0-Type Higher-Order Shear Deformation Theory for Analysis of Laminated Composite Plates

Abstract: The edge-based smoothing discrete shear gap method (ES-DSG3) using three-node triangular elements is combined with a C0-type higher-order shear deformation theory (HSDT) to give a new linear triangular plate element for static, free vibration, and buckling analyses of laminated composite plates. In the ES-DSG3, only the linear approximation is necessary, and the discrete shear gap method (DSG) for triangular plate elements is used to avoid the shear locking and spurious zero energy modes. In addition, the stiffness matrices are calculated relying on smoothing domains associated with the edges of the triangular elements through an edge-based strain smoothing technique. Using the C0-type HSDT, the shear correction factors in the original ES-DSG3 can be removed and replaced by two additional degrees of freedom at each node. The numerical examples demonstrated that the ES-DSG3 show remarkably excellent performance compared to several other published elements in the literature.

Meshless modeling of natural convection problems in non-rectangular cavity using the variational multiscale element free Galerkin method

Abstract: In this paper, the two-dimensional natural convection problems in complex geometries were solved by using the variational multiscale element free Galerkin (VMEFG) method The VMEFG method is a meshless method which coupled element free Galerkin method and variational multiscale method, thus it inherits the advantages of variational multiscale and meshless methods In this method, the field variables are decomposed into coarse and fine scales first, then solved fine scale problem analytically by using bubble functions, in the process, the stabilization parameters had appeared naturally Moreover, it ensures that the resultant formulations yield a consistent stabilized method From the viewpoint of application, the presented method can employ equal order basis for pressure and velocity, which is not only easy to implement but also avoid the restriction of the Babuŝka–Brezzi condition and eliminate non-physical oscillations Several test problems, including natural convection in the semicircular cavity, triangular cavity and triangular cavity with zig-zag shaped bottom wall are considered to investigate the accuracy of the proposed method The numerical results obtained using VMEFG showed very good agreement with those available in the literature

The numerical solution of the two–dimensional sinh-Gordon equation via three meshless methods

Abstract: In this paper three numerical techniques are proposed for solving the nonlinear sinh-Gordon equation. Firstly, we obtain a time discrete scheme then we use the radial basis functions (RBFs) collocation based on Kansa׳s approach, RBF-pseudospectral (PS) technique and moving least squares (MLS) methods to approximate the spatial derivatives. The aim of this paper is to show that the meshless methods based on the RBFs using collocation approach and MLS are suitable for the treatment of the nonlinear partial differential equations and also we compare the mentioned methods in terms of condition number of coefficient matrix and absolute value of error. Also, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

Finite Pointset Method for the Simulation of a Vehicle Travelling Through a Body of Water

Abstract: In order to shorten design cycles and reduce the cost of development in the automotive industry, simulation tools are widely used for analysis and testing throughout the vehicle development process. The Finite Pointset Method (FPM) has been applied to predict numerically the fluid motion as a vehicle travels through a body of water. FPM is a purely meshfree approach based on a generalized finite difference formulation. It discretises the flow field over a cloud of zero-dimensional, numerical points using least squares operators, and is therefore particularly suitable for flows with free surfaces. FPM has already successfully been applied to various industrial problems with significant transient surface deformation, including airbag deployment, glass formation, filling and sloshing processes. Ensuring the ability of a vehicle to travel through a body of water without being damaged is necessary in markets where monsoon rain occurs or where off-road driving is important. The development of the process to use FPM as a simulation tool for water travel was carried out in two stages: a two-phase water/air simulation of an operating air intake over a basin of water, and a single phase simulation with a moving vehicle.

Global Versus Localized RBF Meshless Methods for Solving Incompressible Fluid Flow with Heat Transfer

Abstract: Global and localized radial basis function (RBF) meshless methods are compared for solving viscous incompressible fluid flow with heat transfer using structured multiquadratic RBFs. In the global approach, the collocation is made globally over the whole domain, so the size of the discretization matrices scales as the number of the nodes in the domain. The localized meshless method uses a local collocation defined over a set of overlapping domains of influence. Only small systems of linear equations need to be solved for each node. The computational effort thus grows linearly with the number of nodes—the localized approach is slightly more expensive on serial processors, but is highly parallelizable. Numerical results are presented for three benchmark problems—the lid-driven cavity, natural convection within an enclosure, and forced convective flow over a backward-facing step—and results are compared with the finite-element method (FEM) and experimental data.