1. “ FROM MESH-FREE TO WHERE? ” Development and Progress S. G. AHMED Prof. of Eng. & Applied MathematicsProf. of Eng. & Applied Mathematics President of IEJS SocietyPresident of IEJS Society Director of Computational Mechanics, Zagazig University, EgyptDirector of Computational Mechanics, Zagazig University, Egypt Dept. of Eng. Math & Phys.Dept. of Eng. Math & Phys. Faculty of EngineeringFaculty of Engineering Zagazig University, EgyptZagazig University, Egypt Editor-in-Chief IeJEMTAEditor-in-Chief IeJEMTA Editor-in-Chief IeJNARTEditor-in-Chief IeJNART http://www.iejs.org http://www.iejemta.com http://www.iejnart.com sgahmed911@hotmail.com +2-0106392877

2. Aim & Scope In the past decade several mesh-less techniques have appeared, which are easier to implement continuum problems with free moving boundaries and interfaces. Such problems was difficult to solve with classical techniques as FEM, FVM, FD, because the mesh requires severe modification at each time step. In the present lecture, an overview study for the mesh-free methods will be carried out. This study aims mainly to give the insight to this new branch of numerical methods which still have a lot to do. In this study, a new Radial Basis Function (RBF) is introduced since it developed by the author since 2005 and its results was good compared with other functions.

3. We have new branch of numerical methods This branch, have the ability to solve: (1)A wide range of engineering, applied and industrial problems. (2)Ability to solve problems of large domain, which was impossible to solve in the past. For instance it became possible to simulate large- scale problems such as fluid flow around ships and aircrafts, meteorology, turbulence and wide range of computer intensive problems.

4. WE ARE GOING TOPRESENT

5. (1)MOTIVATIONS Homogeneous and Inhomogeneous partial differential equations are of practical importance, and have a wide range of engineering and industrial applications. With the rapid development of computer technology, the research of the new discipline of engineering mathematics become easily to work and develop new ideas. Since, nearly twenty years ago, new branches of numerical methods started appearing. This branch was so called Mesh-less methods.

6. (2)Historical Background of Radial Basis Functions Since early 90’s, some efficient RBF schemes for numerical PDEs have been presented. Among them are: (1)Kansa method (2)Hermite domain-type RBF collocation method (3)Method of fundamental solution (4)Boundary Knot method ……etc In fact, earlier than Kansa’s pioneer work Nardini and Brebbia in early 80’s without knowing the RBF terminology and existent developments, applied the RBF and the Dual Reciprocity Method to effectively eliminate domain integral in context of the boundary element technique.

7. DUAL RECIPROCITY METHOD · DUAL RECIPROCITY METHOD Stage (1) The dual reciprocity method starts by introducing the following global interpolation form of the body force term b: Where: Interpolation coordinate functions between a point and any point x. Unknown coefficients associated with each of the function. · Stage (2) The next stage is to define the particular solution, corresponding to the function,i.e.

8. This eventually leads to the following expression for the domain integral : After applying, the usual boundary elements discretization equation, gives rise to the following system Where : and are matrices of N x M each for the boundary solutions M is the number of points at which the function has been applied N is the number of boundary unknowns

9. Choice of interpolation Functions The choice of appropriate interpolation functions depends mainly experience. Brebbia and Nardini, proposed functions of the type distance between the points of application of the function and any given point. MULTIPLE RECIPROCITY METHOD At 1989 Nowak introduced what is called the Multiple reciprocity method. Since 1989 Nowak and Brebbia extended and generalized the Concepts of the MRM including Transient and Helmholtz problems. The method interducee a set of so-called higher order fundamental solutions and operates on the domain integral in a recurrent manner. Definition The multiple reciprocity method is technique, which is capable of transforming domain integrals resulting from the generalized body force to the equivalent boundary form.

10. Let us start with the domain integral equation The superscript (0) has been included in the domain integral D as well as the fundamental solution and in generalized body force term b

11. PROCEDURE OF APPLYING MULTIPLE RECIPROCITY METHOD (1) The multiple reciprocity method starts from introducing the so-called fundamental solution of first order and the auxiliary flux. These functions are related to the standard fundamental solution by the formulae Provided these functions are known, the domain integral can be evaluated by making use of the reciprocity theorem :

12. For a known function of space, one can assume that its Laplacian can be derived analytically to obtain a new function, The application of equations (1)and (3) into the equation (2) Results in Where : is the flux analog defined as : The original domain integral has now been transformed into a boundary integral plus a new domain term defined as The procedure described above can be generalized in a natural way. Namely introducing a sequence of higher order fundamental solutions defined by the recurrence formulae:

13. As well as a sequence of body force laplacians · Therefore, the domain integral is transformed into the equivalent Boundary integrals : Finally, introducing equation (22) into equation (4) one arrives at the following boundary only formulation

14. Reciprocity theorem

15. NUMERICAL EXAMPLE POISSON EQUATION The ellipse shown below has a semi-major axis of length 2 and a semi-minor axis of length 1. It is assumed that unit heat conductivity and heat source is proportional to the square of x variable. The governing Poisson equation is : The exact solution which, satisfies homogeneous boundary conditions u=0 on the boundary of the domain shown is given by : In addition, the heat flux on the boundary is then given by:

16. The solution of the present example is based on the boundary nodes only. The heat source function produces two laplacians only, The following table shows the comparision of the multiple reciprocity method (MRM ) analysis against analytical as well as dual reciprocity method (DRM) (58).

17. VariableNodeX coorY coorMRMExactDRM q12.0000.0000.8770.9490.825 21.706-0.5220.9520.9150.948 30.179-0.8080.7110.6570.698 40.598-0.9540.3680.3430.339 50.0000.2280.2080.194 u171.5000.0000.2470.2590.262 181.200-0.3500.2170.220 190.600-0.4500.1500.1430.136 200.000-0.4500.1130.1030.092 210.9000.0000.2440.2400.236 220.3000.0000.1630.1510.142

18. MESH-FREE METHODS…IMPORTANCE Mesh-free method is used to established a system of algebraic equations for the whole problem domain without the use of a pre-defined mesh. Mesh-free methods eliminate some or all of the traditional mesh-based view of the computational domain. A goal of mesh-free methods is to facilitate the simulation of increasingly demanding problems that require the ability to treat large deformations, advanced materials, complex geometry, nonlinear material behavior, discontinuities and singularities.

19. NAMES OF MESH-FREE METHODS The names for the various mesh-free methods are still being debated. Because the methodology is still in a rapid development stage, new names of methods are constantly proposed. It may take some time before all the methods are properly categorized and unified to avoid confusion in the community.

20. MINIMUM AND IDEAL REQUIREMENTS The minimum requirement for mesh-free methods is that a pre-defined mesh is not necessary, at least in the field variable interpolation The ideal requirement for mesh-free methods is that no mesh is necessary at all throughout the process of solving the problem of given arbitrary geometry governed by partial differential system equations subject to all kinds of boundary conditions.

21. IDEA OF MESH-FREE METHODS It is well known for FEM users that creation of a mesh for the problem domain is a pre-requesting in using FEM package. There are many difficulties associated with FEM usage. As long as elements must be used, the problems will not easy to solve. The concept of element-free or mesh-free has been proposed, in which the domain of the problem is represented by a set of arbitrarily distributed nodes. In mesh-free methods; there is no need a priori any information about the relationship of the nodes. This provides flexibility in adding or deleting points/nodes whenever and wherever needed.

22. Geometry Generation FEM Mesh-Free Methods Element Mesh Generation Nodal Mesh Generation Shape Function Creation based on Element Pre-defined Shape Function Creation based on nodes in a local domain System equation for elements System equation for Nodes Global Matrix Assembly Results Assessment FLOW CHART FOR FEM AND MESH-FREE PROCEDURES

23. Conceptual design Simulation Experimental, Analytical and Computational Analysis Design Prototyping Testing Fabrication Modeling Physical, Mathematical, Computational, Operational, Economical The procedure in FEM and Mesh-Free methods for solving engineering problems can be obtained using the following flow chart.

24. Similarities and Differences between FEM and Mesh-Free Methods ItemFEMMesh-Free Element meshYesNo Mesh creation and automation Difficult for 3D caseCan always be done Shape function creationElement basedNode based Shape function propertiesSatisfy Kronecker delta conditions May or may not Satisfy Kronecker delta conditions Discretized system stiffness matrix Bonded, symmetricalBonded, may or may not be symmetric depending on the method used Imposition of essential boundary condition Easy and standardSpecial methods may be required Computation speedFast1.1 to 50 times slower compared to FEM depending on the method used. AccuracyAccurateMore accurate than FEM Commercial software package availability ManyVery few and close to none

25. MESH-FREE METHODS…OVER VIEW Mesh-free methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years mesh-free approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems.

26. Originally, the motivation for two of the most common basic mesh- free approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such as in the numerical solution of PDEs, artificial intelligence, learning theory, neural networks, signal processing, sampling theory and optimization. It should be pointed out that (mesh-free) local regression methods have been used (independently) in statistics for more than 100 years.

27. (1)Smoothed Particle Hydrodynamics (SPH) (2)Diffuse Element Method (DEM) (3)Element-Free Galerkin Method (EFGM) (4)Reproducing Kernel Particle Method (RKPM) (5)Natural Element Method (NEM) (6)Material Point Method (MPM) (7)Mesh-less Local Petrov Galerkin (MLPG) (8)Generalized finite difference method (GFDM) (9)Particle-in-Cell (PIC) (10)Moving Particle Finite Element Method (MPFEM) (11)Finite Cloud Method (FCM) (12)Boundary Node Method (BNM) (13)Boundary Cloud Method (BCM) (14)Method of Finite Spheres (MFS) (15)Moving Least Squares (MLS) (16)Partition of Unity Methods (POUM) AMONG BUT NOT ALL SOME OF THE MOST FAMOUS METHODS

28. SOME OF SHAPE FUNCTIONS USED IN FINITE ELEMENT ANALYSIS finite element analysis The following graphs of shape functions are used to solve problems by finite element analysis. The first three are Langrange functions and the last shows Hermite cubic polynomials. Notice that each function is unity at its own node and zero at the other nodes. Furthermore, the Lagrange shape functions sum to unity everywhere. For the Hermite polynomials H1 and H3 sum to unity.

29. Figure 1. The Lagrange linear shape functions N 1 = 1 - S / L, N 2 = S / L, L = X 2 - X 1, S = X - X 1

30. Figure 2. The Lagrange quadratic shape functions N 1 = (r-1)(2r-1), N 2 = 4r (1 - r), N 3 = r (2r -1)

31. Figure 3. The Lagrange cubic shape functions N 1 = (1 - r) (2 - 3 r) (1 - 3 r) / 2, N 2 = 9 r (1 - r) (2 - 3 r) / 2, N 3 = 9 r (1 - r) (3 r - 1) / 2, N 4 = r (2 - 3 r) (1 - 3 r) / 2

32. Figure 4. The Hermite Cubic Shape functions H 1 = 1 - 3 r 2 + 2 r 3, H 2 = L ( r - 2 r 2 + r 3 ), H 3 = 3 r 2 - 2 r 3, H 4 = L ( - r 2 + r 3 )

33. SHAPE FUNCTION CREATION In the early years of the development of FEM, much of the work involved the formulation of all different types of elements. All the shape functions of finite elements satisfy the Kronecker delta function property. In mesh-free methods, however, the construction of shape functions has been and still is the central issue. In mesh-free methods, the shape function changes as the location of the point of interest changes. The procedure in FEM and Mesh-Free methods for solving engineering problems can be obtained using the following flow chart.

34. PROPERTIES OF SHAPE FUNCTION (1)Shape function must satisfy the partition of unity condition, that is necessary to produce any rigid motion of the problem domain (2)Linear field reproduction condition This condition is required for the shape function to pass the standard patch test in FEM but Not necessary when using mesh-free methods (3) Kronecker delta function property

35. MESH-FREE METHODS PROCEDURE (1)Domain representation: The solid body of the structure is first modeled and is represented using sets of nodes scattered in the problem domain and its boundary. (2)Displacement interpolation: The field variable at any point within the problem domain is interpolated using the displacement and its nodes within the support domain.

36. Where n:Number of nodes included in “ small local domain ” :Nodal field variable :Vector that collects all the field variables at these nodes. Small local domain is termed as the supported domain of x. Support domain of a point x determines the number of nodes to be used to support or approximate the function value at the point x.

37. (3)Formation of system of equations The global system equations are: A set of algebraic equations for static analysis Eigenvalue equations for free-vibration analysis Differential equations w.r.t. time for general dynamic problems. (4)Solving the global mesh-free equations The numerical method used depends mainly on the problem solved, static problem, free- vibration problem or dynamics problems

38. RADIAL BASIS FUNCTION A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that ; or alternatively on the distance from some other point c, called a center, so that. Any function φ that satisfies the property φ(x)=φ(||x||) is a radial function. The norm is usually Euclidean distance. Radial basis functions are typically used to build up function approximations of the form. where the approximating function y(x) is represented as a sum of N radial basis functions, each associated with a different center c i, and weighted by an appropriate coefficient w i. Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour.

39. SOME OF CLASSICAL RBF AND NEW ONE Gaussian ( GA ) (2)Multiqudric ( MQ ) (3)Logarithmic (4) Thin plate (TP ) (5)S. G. Ahmed’s function The last RBF based on a collocation technique was developed and tested since 2005 by Ahmed, S. G. Likely this function with the suggested technique gave an excellent results compared with the previous ones. The details of that paper will be explained in more details as follow.

40. PAPER PUBLISHED BY S. G. AHMED Engineering Analysis with Boundary Elements 2005 A COLLOCATION METHOD USING NEW COMBINED RADIAL BASIS FUNCTIONS OF THIN PLATE AND MULTI-QUADRAIC TYPES In the present paper, a mesh-less method based on a new combination between thin plate and multi-quadraic radial basis functions is developed. The new form of the radial basis function contains a parameter, which named a control parameter. Starting the proposed method by the classical weight implicit finite difference approximation, then fellow up the same procedure as thin plate or multi-quadraic but with the new form of the radial basis function. The control parameter, will take ONE when evaluating the diagonal elements and ZERO for the coefficients matrix resulted.

41. THE PROPOSED METHOD Consider, for brief explaination, a three-dimensional heat transfer problem defined as with Starting with implicit scheme using Therefore equation (1) will be of the form:

42. Re-arrange and simplify, we get where assume in the present paper Where

43. And is a control parameter, is a constant shape parameter is a constant equals (2) in case of heat conduction problems Now for any linear partial differentiation, can be approximated by using the concept of equation (12) into equation (5), we get

44. let for simplicity Therefore; equation (13) will be where

45. Therefore, equation (16) can be re-cast in the following matrix form

46. Proposed Algorithm The starting time step solution comes from applying equation (8) at the selected points inside the domain. Re-arrange the system in a matrix form will take the following form This system can be re-written as Their solution will be The successive time steps solutions come from solving system of linear equations given by equation (19), which written as

47. let Therefore, system (23) can be re-written as Their solution takes the form

48. Numerical Results Example (1) Consider a one-dimensional heat conduction with Dirichlet condition, defined as with The analytical solution as given in Carslaw and Jaeger With the following numerical data

49. In this example, the diagonal elements of the systems given by equations (22) and (26) are evaluated corresponding to and the other elements are evaluated corresponding to. A comparison between the present method and the analytical solution of the problem is made as shown in Table 1. In this table, for every time the first and the second row corresponding to the analytical and the present results respectively. The third row at each time corresponds to the relative error between the results of the present method and the analytical solution. On the other- hand the relative errors at different points was found in the range, Also it can be seen that ant any time step, a symmetrical temperature distribution around the mid-way between the two ends of the domain is obtained with a very small error due to approximation.

50. Time 0.1Analytical0.1996100.5798980.7743240.5798980.199610 Present0.1996250.5799040.7743250.5799060.199622 R.E. 0.3Analytical0.1812350.4857490.6091280.4857490.181234 Present0.1812470.4857840.6091280.4857760.181240 R.E. 0.5Analytical0.1520590.4000150.4959120.4000150.152059 Present0.1520630.4959140.4000370.152068 R.E. 0.7Analytical0.1253800.3285730.4063870.3285730.125380 Present0.1253870.3285910.4063880.3286030.125392 R.E. 0.9Analytical0.1030150.2697520.3334740.2697520.103015 Present0.1031400.2697600.3334750.2697670.103029 R.E.

51. Example (2) Consider a two-dimensional diffusion problem over a square slab of dimension, with and. The associated initial and boundary conditions are as follow: In this example, two different cases for the control parameter, were done. In the first case we used two values of the control parameter when evaluating matrix coefficients. We took for all elements and took for elements, while in the second case we took constant value for, which practically corresponding to the average between the thin plate and the multi-quadraic functions.

52. The comparison between the present method and the analytical solution is made as shown in Figure 1 at different times. It is clear from the graph that the error between the first case solution and the analytical solution at the same time step is small and can be neglected. Meanwhile, the error between the second case of the control parameter and the analytical solution is still small and can be improved by trying different values for the control parameter, or the shape parameter c or also by increasing the scattared points inside the domain.

53. Figure 1: Results of example (2) at different times

54. CONCLUSION Thin plate and multi-quadraic meshless methods have been applied to a wide range of engineering problems, each method has advantages and disadvantages. When dealing with the multi-quadraic the main disadvantage is how to choose the shape parameter, c, specially some researchers found that it is a problem dependent while others found that by varying it the accuracy of the results improved. The multi-quadraic has the advantage that no singularities appear when computing the diagonal elements of the system of the linear equations obtained. On the other hand the thin plate has the advantage that there is no shape parameter, therefore, has no effect but it has the disadvantage of the singularity when evaluating the afro-mentioned diagonal elements. The present paper took the advantages of both methods and overcome the major disadvantages of them. Also the present method has what is called a control parameter which gives the facility of the researchers to try different values of and subsequently a major band of error analysis.

55. TO WHERE WE ARE GOING TOWARDS ? Due to the advances progress in modeling and simulation of practical and industrial problems and due to rapid development of computer technologies, the need to quick and accurate numerical methods and results became the main direction of most research. I hope everyone interesting dealing with Numerical Analysis think a lot to start with this new branch of Numerical Analysis.