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## Advances in Computational Mechanics

Vol.1, 2009
The MLPG Meshless Methods |

Volume Editor: Jan Sladek |

## CHAPTER ITHE MLPG METHOD FOR THE SOLUTION OF THE HEAT EQUATION WITH MOVING BOUNDARIES Rubén Avila Abstract The solution of phase change (solid-liquid) problems in which a moving interface is present, has been traditionally obtained by using mesh based methods. However when the deformation of the interface is severe, a high computer cost remeshing process must be performed. In this investigation, the MLPG method has been used to solve the nonsteady heat diffusion equations that govern the melting process in systems where the natural convection of the liquid phase is neglected. The heat equations are formulated by using both: (i) an Eulerian frame of reference (for the nodes located far from the interface) and (ii) an Arbitrary Lagrangian Eulerian (ALE) approach (for the nodes located close to the interface). In order to obtain the displacement of the nodes located at the interface, we solve, by using the MLPG method, the heat balance equation at the moving boundary. To update the position of the nodes used to discretize the domain we solve (by using the MLPG method) an elliptic elasto-static model which is based on the solution of a Poisson equation. The temporal discretization of the heat equations is carried out by using the Crank-Nicolson scheme. The Moving Least Squares (MLS) scheme is used to generate the interpolation functions. The integration of the governing equations is performed by using the Gauss-Lobatto-Legendre quadrature rule. The weight function used in the MLS scheme and in the weighted residual process, is a compact support fourth order spline. The numerical simulation of three cases is presented: (i) melting process of a solid bar, (ii) melting process of an annular solid conﬁned between concentric circles and (iii) melting process of an annular solid conﬁned between concentric spheres. The results obtained by the MLPG method have been compared with the numerical solution provided by the mesh based Spectral Element Method. It is concluded that the MLPG method can be used as a reliable methodology for the solution of the heat equations with a moving boundary. Keywords: Meshless Methods, Phase Change, Heat transfer, Moving Least Squares. 1 Departamento de Termoﬂuidos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Mexico D.F., C.P. 04510, ravila@servidor.unam.mx., Center for Aerospace Research & Education, University of California, Irvine. 2 Center for Aerospace Research & Education, University of California, Irvine. ## CHAPTER IITHE MLPG MIXED COLLOCATION METHOD FOR MATERIAL ORIENTATION AND TOPOLOGY OPTIMIZATION OF ANISOTROPIC SOLIDS AND STRUCTURES Shu Li Abstract In this paper, a method based on a combination of an optimization of directions of orthotropy, along with topology optimization, is applied to continuum orthotropic solids with the objective of minimizing their compliance. The spatial discretization algorithm is the so called Meshless Local Petrov-Galerkin (MLPG) “mixed collocation” method for the design domain, and the material-orthotropy orientation angles and the nodal volume fractions are used as the design variables in material optimization and topology optimization, respectively. Filtering after each iteration diminishes the checkerboard effect in the topology optimization problem. The example results are provided to illustrate the effects of the orthotropic material characteristics in structural topology-optimization. Keywords: orthotropy, material-axes orientation optimization, topology optimization, meshless method, MLPG, collocation, mixed method 1 Department of Aircraft Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China 2 Center for Aerospace Research & Education, University of California, Irvine, USA ## CHAPTER IIIA STUDY OF BOUNDARY CONDITIONS IN THE MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD FOR ELECTROMAGNETIC FIELD COMPUTATIONS Meiling Zhao Abstract Meshless local Petrov-Galerkin (MLPG) method is successfully applied for electromagnetic ﬁeld computations. The moving least square technique is used to interpolate the trial and test functions. More attention is paid to imposing the essential boundary conditions of electromagnetic equations. A new coupled mesh-less local Petrov-Galerkin and ﬁnite element (MLPG-FE) method is presented to enforce the essential boundary conditions. Unlike the conventional coupled technique, this approach can ensure the smooth blending of the potential variables as well as their derivatives in the transition region between the meshless and ﬁnite element domains. Then the boundary singular weight method is proposed to enforce the boundary conditions for electromagnetic ﬁeld equations accurately. Practical examples in engineering, including the computations of the electric-ﬁeld intensity of the cross section of long straight metal slot, the end region of a power transformer and axisymmetric problem in the electromagnetic ﬁeld, are solved by the presented approaches. All numerical veriﬁcation and all kinds of comparison analysis show that the MLPG method is a promising alternative numerical ap 1 Department of Applied Mathematics, School of Mathematics and Systems Science, Beijing University of Aeronautics & Astronautics, Beijing, 100083, P. R. China 2 Department of Applied Mathematics, Northwestern Polytechnic University, Xi’an, Shaanxi, 710072, P. R. China ## CHAPTER IVMODELING OF PIEZOELECTRIC AND PIEZOMAGNETIC SOLIDS BY THE MLPG J. Sladek Abstract A meshless method based on the local Petrov-Galerkin approach is proposed to solve 2-D and 3-D axisymmetric boundary value problems in piezoelectric and magneto-electric-elastic solids with continuously varying material properties. Axial symmetry of geometry and boundary conditions reduces the original 3-D boundary value problem into a 2-D problem in the axial cross section. Stationary and transient dynamic problems are considered in this paper. The mechanical ﬁelds are described by the equations of motion with an inertial term. To eliminate the time-dependence in the governing partial differential equations the Laplacetransform technique is applied to the governing equations, which are satisﬁed in the Laplace-transformed domain in a weak-form on small subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variation of the displacements and the electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of linear algebraic equations for unknown nodal values. The boundary conditions on the global boundary are satisﬁed by the collocation of the MLS-approximation expressions for the displacements and the electric potential at the boundary nodal points. The Stehfest’s inversion method is applied to obtain the ﬁnal time-dependent solutions. 1 Institution of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia, sladek@savba.sk 2 Department of Mechanics, Slovak Technical University, Bratislava, Slovakia 3 Department of Mechanical and Aerospace Engineering, University of California, Irvine, USA ## CHAPTER VANALYSIS OF HEAT CONDUCTION PROBLEMS IN 3D ANISOTROPIC FUNCTIONALLY GRADED SOLIDS, BY THE MESHLESS LOCAL PETROV-GALERKIN (MLPG) METHOD J. Sladek Abstract A meshless method based on the local Petrov-Galerkin approach is proposed, for the solution of steady-state and transient heat conduction problems in a continuously non-homogeneous anisotropic medium. Both the Laplace transform and the time difference approaches are used to treat the time dependence of the variables for transient problems. The analyzed domain is covered by small subdomains with a simple geometry. A weak formulation for the set of governing equations is transformed into local integral equations over local subdomains, by using a unit test function. Nodal points are randomly distributed in the 3D analyzed domain, and each node is surrounded by a spherical subdomain to which a local integral equation is applied. Spatial variation of the temperature and heat ﬂux (or of their Laplace transforms) at discrete time instants are approximated on the local boundary and in the interior of the subdomain, by means of the moving least-squares (MLS) method. The Stehfest algorithm is applied for the numerical Laplace inversion, in order to retrieve the time-dependent solutions. 1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia 2 Department of Mechanical & Aerospace Engineering, Carleton University, Ottawa, Canada K1S 5B6 3 Center of Aerospace Research & Education, University of California at Irvine, Irvine, CA 926973975, USA Keywords: meshless method, local weak form, Heaviside step function, moving least squares interpolation, Laplace transform, time difference approach ## CHAPTER VIA FINITE VOLUME MESHLESS LOCAL PETROV-GALERKIN METHOD FOR TOPOLOGY OPTIMIZATION DESIGN OF THE CONTINUUM STRUCTURES Zheng Juan Abstract In this paper, the ﬁnite volume meshless local Petrov-Galerkin method (FVMLPG) is applied to carry out a topology optimization design of the continuum structures. In FVMLPG method, the ﬁnite volume method is combined with the meshless local Petrov-Galerkin method, and both strains as well as displacements are interpolated, at randomly distributed points in a local domain, using the moving least squares (MLS) approximation. The nodal values of strains are expressed in terms of the independently interpolated nodal values of displacements, by simple enforcing the strain-displacement relationships directly. Considering the relative density of nodes as design variable, and the minimization of compliance as objective function, the mathematical formulation of the topology optimization design is developed using the solid isotropic microstructures with penalization (SIMP) interpolation scheme. The topology optimization problem is solved by the optimality criteria method. Numerical examples show that the proposed approach is feasible and efﬁcient for the topology optimization design of the continuum structures. 1State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, China 2Corresponding author. Tel.: +86-0731-8824724. E-mail: dingdang8209@163.com 3College of Mechanics and Aerospace Engineering, Hunan University, Changsha, China Keywords: ﬁnite volume meshless local Petrov-Galerkin method (FVMLPG); moving least squares (MLS); topology optimization design for continuum structures; SIMP; optimality criteria method ## CHAPTER VIIA MESHLESS METHOD BASED ON A COMBINED FORMULATION OF MFVM AND MCM FOR ELASTO-STATIC PROBLEMS Y. H. Xiao Abstract A new meshless method is developed for solving elasto-static problems based on a combined discretization approach of Meshless Finite Volume Method (MFVM) and Meshless Collocation Method (MCM). In this method, the problem domain and its boundaries are represented by a set of scattered nodes. These nodes are categorized into two types, boundary node and internal node. To establish the discrete system equations, two different formulations are used for these two types of node, respectively. The MFVM is used for boundary nodes so that natural boundary conditions can be satisﬁed naturally. The MCM is used for internal nodes, so no numerical integration is required for these nodes. The moving least square (MLS) method is employed for interpolation and the penalty method is applied to impose essential boundary conditions in the present method. Considering that the MCM involves evaluating the second derivatives of displacements, which is a time consuming task for MLS, a method based on two sequential ﬁrst derivative approximations is proposed to overcome this problem. Several numerical examples are included to illustrate the efﬁciency, accuracy and convergence of the proposed method. The results show that high accuracy and good rate of convergence can be obtained and the computational efﬁciency is improved evidently compared with the MFVM. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, China E-mail address: xyhtome@yahoo.com.cn (Y. H. Xiao) hud_a@163.com (D. A. Hu) hanxu@hnu.cn (X. Han) Keywords: meshless method; Meshless Finite Volume Method; Meshles Collocation Method; moving least square; second order derivatives
EXPERIMENTAL ANALYSIS OF METHODS FOR MOVING LEAST SQUARES SUPPORT DETERMINATION Roman Trobec Abstract The moving least squares (MLS) is an essential approach that guarantees the locality of many meshless methods for the numerical solution of partial differential equations (PDE). The MLS is an extension of the classical least squares in the way that just a few nearest discretization points have impact on the approximated value in a selected point. The locality is implemented by hat-shaped MLS weight functions applied on local support domains. The character of weight functions and the dimension of MLS support domains have a signiﬁcant impact on the accuracy of the MLS approximation, particularly in case of non-uniform distribution of points. In meshless local Petrov-Galerkin method with MLS weight functions as test functions (MLPG1), MLS weight functions also determine the quadrature domains and therefore have an important impact on the calculation complexity and accuracy of MLPG1 solutions. We conﬁrmed with experimental analysis that there is only a short interval of MLS support radii that provides acceptable MLPG1 solutions. Three methods for the determination of the MLS support domain have been analyzed experimentally: constant radius, sampled and interpolated radius and continuously variable radius with selected numbers of support points. Several testing conditions have been proposed for the uniﬁcation of the testing methodology. Systems with uniform and non-uniform point distributions have been tested. The MLS approximation errors and MLPG1 solution errors were analyzed. Our results Institut Jožef Stefan, Jamova 39, 1000 Ljubljana, Slovenia, roman.trobec@ijs.si could contribute to the future theoretical development and research of the stability and convergence of the MLPG methods. Keywords: moving least squares approximation, MLS, support domain, mesh-less local Petrov-Galerkin, MLPG, PDE, non-uniform points, nodes.
IMPROVING THE CONVERGENCE OF THE HERMITE CVRBF METHOD FOR UNSTRUCTURED MESHES P. Orsini Abstract In this work two alternatives to improve the convergence of the Hermite Control Volume Radial Basis Function (HCVRBF) method for unstructured meshes are investigated: Increasing the order of the numerical integration schemes, and the use of vertex centred (VC) discretization, which guarantees a numerically conservative scheme. The convergence and the conservation of the ﬂux analysis of these two approaches implemented to improve the performance of the HCVRBF method are carried out in a one-dimensional advection diffusion problem using three unstructured meshes progressively reﬁned. In addition, the numerical behaviour of these approaches is also assessed for steady and unsteady three dimensional advection diffusion problems using unstructured meshes. The University of Nottingham, School of Mechanical, Materials and Manufacturing Engineering, Nottingham, NG7 2RD, United Kingdom. Correspondence to: H. Power, henry.power@nottingham.ac.uk |