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where W is the diagonal matrix constructed using a weight function, 1.e.,

Figure 8 where W is the diagonal matrix constructed using a weight function, 1.e.,

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Figure 3.2. Pascal triangle of monomials for two-dimensional domains.
Figure 3.2. Pascal triangle of monomials for two-dimensional domains.
O Interior nodes and nodes on the Dirichlet boundary ® Collocation node @ Nodes on Derivative boundary (DB-nodes)
O Interior nodes and nodes on the Dirichlet boundary ® Collocation node @ Nodes on Derivative boundary (DB-nodes)
Table 3.2 Typical radial basis functions with dimensionless shape parameters q open to any real variable, and found that g=0.98 or 1.03 led to good results in the analysis of two-dimensional solid and fluid mechanics problems. This will be investigated further in Chapter 4 and Chapter 5.
Table 3.2 Typical radial basis functions with dimensionless shape parameters q open to any real variable, and found that g=0.98 or 1.03 led to good results in the analysis of two-dimensional solid and fluid mechanics problems. This will be investigated further in Chapter 4 and Chapter 5.
able 3.3. Formulations of the compactly supported radial basis function (CSRBF)
able 3.3. Formulations of the compactly supported radial basis function (CSRBF)
where G is the generalized moment matrix that consists of: Arranging Equations (3.98)~(3.100) together leads to the following set of equations in matrix form.
where G is the generalized moment matrix that consists of: Arranging Equations (3.98)~(3.100) together leads to the following set of equations in matrix form.
and the moment matrix of the Ist derivatives of polynomials evaluated at npg DB-nodes, the moment matrix of RBFs evaluated at n nodes,
and the moment matrix of the Ist derivatives of polynomials evaluated at npg DB-nodes, the moment matrix of RBFs evaluated at n nodes,
the moment matrix of lst normal derivatives of RBFs evaluated at the DB- nodes,
the moment matrix of lst normal derivatives of RBFs evaluated at the DB- nodes,
Finally, the approximated function and its derivatives can be obtained using Equation (3.111).
Finally, the approximated function and its derivatives can be obtained using Equation (3.111).
Therefore, the derivatives of the RPIM shape functions can also be obtained by solving Equations (3.120) and (3.121) using the standard linear equation solver.
Therefore, the derivatives of the RPIM shape functions can also be obtained by solving Equations (3.120) and (3.121) using the standard linear equation solver.
Figure 3.11. RPIM shape functions for the node 13 at x'=[0,0] along the line of y=0 obtained using different RBFs.
Figure 3.11. RPIM shape functions for the node 13 at x'=[0,0] along the line of y=0 obtained using different RBFs.
Figure 3.16. The approximate function u"(x) and the nodal parameters u; in the MLS approximation.
Figure 3.16. The approximate function u"(x) and the nodal parameters u; in the MLS approximation.
Note that the WLS formulation mentioned in Sub-section 3.2.1.2 is ver similar to the MLS formulation. In the MLS, the coefficient a is the functio of x which makes the approximation of weighted least squares mov continuously. Therefore, the MLS shape function will be continuous in th entire global domain, as long as the weight functions are chosen properly This global continuity feature is preferred in the MFree global weak-fon methods (Chapter 4). In WLS, however, because the coefficient a i Equation (3.32) is the constant, the WLS shape functions are piecewis continuous, as discussed in Section 3.2.1.2. The WLS approximation can b viewed as a special form of the MLS approximation.
Note that the WLS formulation mentioned in Sub-section 3.2.1.2 is ver similar to the MLS formulation. In the MLS, the coefficient a is the functio of x which makes the approximation of weighted least squares mov continuously. Therefore, the MLS shape function will be continuous in th entire global domain, as long as the weight functions are chosen properly This global continuity feature is preferred in the MFree global weak-fon methods (Chapter 4). In WLS, however, because the coefficient a i Equation (3.32) is the constant, the WLS shape functions are piecewis continuous, as discussed in Section 3.2.1.2. The WLS approximation can b viewed as a special form of the MLS approximation.
In the MLS approximation, a support domain, defined in Equation (3.1), can be formed for any point of interest. Field nodes included in this support domain are used to perform the MLS approximation for the unknown function at this point. The number of nodes, 1, chosen in the support domain, should be sufficient to ensure that the matrix A in Equation (3.134) is invertible, so as to provide the interpolation stability (Condition 2 in Sub-section 3.1.1). The choice of m depends on the nodal distribution and the number of basis functions, m. In order to ensure the existence of A”! and a well-conditioned A, we usually let 1 >>m. Unfortunately, there is no theoretical best value of n, and it has to be determined by numerical experiments.
In the MLS approximation, a support domain, defined in Equation (3.1), can be formed for any point of interest. Field nodes included in this support domain are used to perform the MLS approximation for the unknown function at this point. The number of nodes, 1, chosen in the support domain, should be sufficient to ensure that the matrix A in Equation (3.134) is invertible, so as to provide the interpolation stability (Condition 2 in Sub-section 3.1.1). The choice of m depends on the nodal distribution and the number of basis functions, m. In order to ensure the existence of A”! and a well-conditioned A, we usually let 1 >>m. Unfortunately, there is no theoretical best value of n, and it has to be determined by numerical experiments.
Using Equations (3.154), (3.155) and (3.156), we can obtain the following set of equations. Solving the above set of equations for b; yields
Using Equations (3.154), (3.155) and (3.156), we can obtain the following set of equations. Solving the above set of equations for b; yields
3. Meshfree shape function construction
3. Meshfree shape function construction
When the quartic spline weight function (W2) is used, we have
When the quartic spline weight function (W2) is used, we have
Figure 3.20. MLS shape function for node 13 at x'=[0, 0] obtained using 2: nodes shown in Figure 3.5. First, by adding up the values of ¢; at all the 25 nodes, we can confirm the fact that the MLS shape function is of a partition of unity, i.e.
Figure 3.20. MLS shape function for node 13 at x'=[0, 0] obtained using 2: nodes shown in Figure 3.5. First, by adding up the values of ¢; at all the 25 nodes, we can confirm the fact that the MLS shape function is of a partition of unity, i.e.
Figure 3.23. Effects of weight functions on the MLS shape function of node 13 and its derivative. The results are plotted along the line of y=0 using different weight functions. (a) the shape function; (b) the first-order derivative of the shape function.
Figure 3.23. Effects of weight functions on the MLS shape function of node 13 and its derivative. The results are plotted along the line of y=0 using different weight functions. (a) the shape function; (b) the first-order derivative of the shape function.
It should be mentioned here that the interpolation error is only one part of total error in an MFree method in solving a problem of computational mechanics. The studies of shape parameters presented in this section are only for checking the interpolation quality and the producibility of MLS and RPIM shape functions. The accuracy will be also studied in the following chapters in the analysing actual problems of computational mechanics. chapters in the analysing actual problems of computational mechanics.
It should be mentioned here that the interpolation error is only one part of total error in an MFree method in solving a problem of computational mechanics. The studies of shape parameters presented in this section are only for checking the interpolation quality and the producibility of MLS and RPIM shape functions. The accuracy will be also studied in the following chapters in the analysing actual problems of computational mechanics. chapters in the analysing actual problems of computational mechanics.
Appendix 3.2. Dummy arguments used in the subroutine RPIM_ShapeFunc_2D
Appendix 3.2. Dummy arguments used in the subroutine RPIM_ShapeFunc_2D
Appendix 3.3. Dummy arguments used in the subroutine Compute_RadialBasis Appendix 3.3. Dummy arguments used in the subroutine Compute_RadialBasis Appendix 3.4. Dummy arguments used in the subroutine GaussEqSolver Sym
Appendix 3.3. Dummy arguments used in the subroutine Compute_RadialBasis Appendix 3.3. Dummy arguments used in the subroutine Compute_RadialBasis Appendix 3.4. Dummy arguments used in the subroutine GaussEqSolver Sym
Appendix 3.7. Subroutines used in the program of MLS_ShapeFunc_2D.f9(
Appendix 3.7. Subroutines used in the program of MLS_ShapeFunc_2D.f9(
nodes shown in Figure 3.5, and weight function W1 and mm=3*
nodes shown in Figure 3.5, and weight function W1 and mm=3*
dimension gpos (nx) ,x(nx,numnode) , nv (numnode) , ds (nx, numnode) dimension a(mm,mm,10),b(mm, ndex,10) dimension xv(nx,ndex) aif (nx, ndex) , w(ndex,10),p(6,ndex) ,pp (mm,mm
dimension gpos (nx) ,x(nx,numnode) , nv (numnode) , ds (nx, numnode) dimension a(mm,mm,10),b(mm, ndex,10) dimension xv(nx,ndex) aif (nx, ndex) , w(ndex,10),p(6,ndex) ,pp (mm,mm
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