Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

Analytical particular solutions of the polyharmonic multiquadrics are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.


Introduction
Boundary-type numerical methods have been emerged as a popular research field because only boundary discretizations are required when they are applied to solve homogeneous problems.These methods include the boundary element method (BEM) [1], method of fundamental solutions (MFS) [2,3], and Trefftz methods (TM) [4,5].When they are applied for solving an inhomogeneous partial differential equation, the dual reciprocity method (DRM) can be used [6,7].
In the early development of DRM, the ad hoc radial function, 1 + , was exclusively used.In order to improve the accuracy of the computation, Golberg and Chen [8] and Karur and Ramachandran [9] applied the theory of radial basis functions (RBFs) to the DRM.Among these RBFs, Hardy's multiquadrics (MQ) [10] and Duchon's augmented polyharmonic spline (APS) [11] are the most popular ones.For example, Golberg [12,13], Chen [14], and Karur and Ramachandran [9] demonstrated the superiority of the APS over the ad hoc radial function.Then, Golberg et al. [15] further improved the accuracy of the approximated particular solution by utilizing the MQ.Recently, Tsai [16] generalized the MQ to the polyharmonic multiquadrics (PMQ) and showed further accuracy improvement of the PMQ over the traditional MQ.In this study, the accuracy improvement of the PMQ over the MQ and APS is demonstrated for the Reissner and Mindlin thick-plate models.
When the MQ is adopted in a DRM procedure, the applicability depends on the availability of the analytical particular solution of the basis function associated with the partial differential operator of a given problem.Golberg et al. [15] derived the analytical particular solutions for the Laplace equation.Samaan and Rashed [17,18] and Tsai and Hsu [19] found the analytical particular solutions, respectively, for the two-and three-dimensional elasticity problems, which can be converted to a biharmonic equation using the Galerkin-Papkovich vector [20].Basically, the applicability of MQ is limited to the harmonic and biharmonic operators [21] before Tsai [16] derived the analytical particular solutions of MQ associated with the polyharmonic operators, which are denoted as the PMQ in the following.
When applying the DRM for solving the Mindlin thickplate model [22], Wen et al. [23] and Tsai and Wu [24] found the analytical particular solutions for the ad hoc RBF and the APS, respectively.In addition, the analytical particular solutions for the Chebyshev polynomials [25] and the APS [26] were derived for the Reissner thick-plate model [27,28].Basically, a product operator of biharmonic and Helmholtz 2 Mathematical Problems in Engineering operators resulted when applying the Hörmander operator decomposition technique [29] to the coupled PDEs of the Reissner and Mindlin thick-plate models as demonstrated in the previous studies [24][25][26].This seems to infer that the accuracy improvement of the PMQ over the APS is not possible for the thick-plate models.
In this study, a method is introduced to eliminate the Helmholtz operator, which makes the application of the PMQ for the thick-plate models possible.Numerical experiments are carried out to demonstrate the superior accuracy of the PMQ over the traditional MQ and the first few orders of the APS.
This paper is organized as follows: the unified Reissner/Mindlin plate model is introduced in Section 2.Then, a review of the PMQ is given in Section 3.And the particular solutions of the PMQ associated with the Reissner/Mindlin plate model are derived in Section 4. Section 5 gives the formulation of MFS-DRM.Finally, some numerical experiments are carried out to validate the particular solutions in Section 6 and the conclusions are drawn in Section 7.

Reissner/Mindlin Plate Model
In the following, indices , ,  are in the range {1, 2, 3} and indices , ,  are in the range {1, 2}.As described in Figure 1, we consider a plate of uniform thickness ℎ with its middle plane being a domain Ω with boundary Γ in the  1 −  2 plane and thickness coordinate  3 .The plate is subjected to a normal loading with intensity ( 1 ,  2 ).Then, the equations of equilibrium are given by where   is the transverse shear force and Υ  is the bending or twisting moment.
In the Reissner [27,28] and Mindlin [22] thick-plate theory,   represents an average slope angle and  3 is the lateral deflection of the plate in the middle surface.According to the theories, the constitutive equations are given by for the Mindlin plate model and for the Reissner plate model.Combining (1)∼(2) results in the following governing equation: with ) ) ) ) , (6) where  and V are Young's modulus and Poisson ratio, respectively.
Equation ( 5) is a coupled system of three second-order PDEs for the three unknown functions   , and, therefore, three boundary conditions are required as where   ( 1 ,  2 ) are given boundary data.

Polyharmonic Multiquadrics
Before deriving the particular solutions of PMQ associated with the Reissner/Mindlin plate model, we give a short review on the PMQ previously derived by the method of undetermined coefficients together with the Laurent series in [16].Considering the th order PMQ   , it is governed by the following hierarchical relation: = 0 as  → 0 for 0 ≤  ≤  − 1.
Here, (10) is introduced so that the PMQ are hierarchically unique and  is a shape parameter to tune the numerical accuracy [10].Then, by using the method of undetermined coefficients, we can find the solution of (8)∼ (10) as with for any positive integer .And, the unknown  , in (11) can be solved by using the uniqueness condition (10) as with In ( 14), the double factorial is defined as The PMQ defined in (11) can be expanded into the Maclaurin series as Observing ( 16), we can conclude that the proposed PMQ are infinitely differentiable since they only consist of evencoefficient polynomials [16,30].Then, substituting ( 16) into ( 8) and ( 9) and performing some mathematical manipulations can result in which can be used for solving the unknown coefficients  , and  , in ( 11) and (16).In practice, we simply enforce  equal to 0, 1, . . ., 2 in (17) which results in 2 + 1 linear equations for solving the 2 + 1 unknown coefficients  , and  , of the PMQ.After  , and  , are obtained, (13) can be used to have  , .Then, the PMQ   can be computed simply by (11).For the cases when  is very small or equal to zero, one can alternatively use the Maclaurin series (16).
For the purpose of a later derivation of the particular solutions associated with the Reissner/Mindlin plate model, we need to find the formulas when the radial differential operator (/)  is applied onto (11) and ( 16).This can be done for arbitrary nonnegative integer  since the proposed PMQ are infinitely differentiable.Then, the required formulas for ( 11) and ( 16) can be derived, respectively, as , () , () In deriving (18), we have used And, in deriving ( 20) and ( 21), we have used the product rule of differentiation as follows: with  and  being two arbitrary radial functions.Finally, additional formulas are required to complete ( 20) and ( 21), respectively, as follows: with The proposed formulas can be implemented into a floating-point subroutine for evaluating (/)    () with arbitrary nonnegative integers  and .Therefore, the subroutine can be invoked when implementing numerical methods.In addition, the source codes can be obtained from the author via e-mails.

Analytical Particular Solution of Plate Model
Analytical particular solutions are required when the DRM is applied for solving a Reissner/Mindlin plate problem under an arbitrary loading.In this section, the analytical particular solutions of the PMQ associated with the Reissner/Mindlin plate model will be derived, which are governed by where Formally, the particular solutions   ( 1 ,  2 ), governed by (27), should be obtainable by using the Hörmander operator decomposition technology [29], which begins with the definition of the adjoint operator Ladj as where det( L) is the determinant of L and Ĩ is the threeby-three identity matrix.Here, both det( L) and Ladj can be obtained by linear algebraic manipulations, respectively, defined as ) ) ) .
Then, the particular solutions can be found by assuming where () is an unknown function to be determined.In (32), the required operator Ladj Ẽ can be obtained as After substituting (32) into (27) and using ( 29) and ( 30), the resulted equation indicates that the following particular solution is required: However, the analytical particular solution of the above equation is not known due to the existence of the Helmholtz operator.
In this study, we find that the Helmholtz operator can be eliminated by rewriting (33) as where the operator Ẽ1 is defined by ) .(36) Then, ( 29) and ( 35) can be combined to have The above equations have suggested us to modify the Hörmander operator decomposition technology by assuming where  1 () is an unknown function to be determined.Substituting (38) into (27) and using (37) result in By using the PMQ definitions ( 8) and ( 9), we can have the particular solution of (39) as Substituting (40) into (38) and using (36) can result in the desired analytical particular solutions of displacements as where we have used In addition, the required formulas of  +1 / and  +2 /  have been provided in the previous section.Then, the corresponding shear forces and bending/twisting moments can be obtained by using the definitions Equations ( 41), ( 42), (44), and (45) are sufficient to obtain the analytical particular solutions of shear forces and bending/twisting moments, respectively, as follows: where the additional terms (/) 2  +1 and (/) 2  +2 have been given in the previous section.This completes the derivation of the analytical particular solutions of the PMQ associated with the Reissner/Mindlin plate model.

MFS-DRM Formulation
Now, we are in a position to review the application of the MFS-DRM procedure [24,26] for solving the well-posed thick-plate problem governed by ( 5) and (7).First of all, we need the principle of superposition as where the particular solution    satisfies without specifying any boundary condition.In addition, the homogeneous solution   ℎ satisfies with modified boundary conditions where u ℎ = ( 1 ℎ ,  2 ℎ ,  3 ℎ ) and u  = ( 1  ,  2  ,  3  ).In order to apply the DRM for solving the particular solution    , the loading intensity needs to be first approximated by the PMQ as follows: where   is the distance between ( 1 ,  2 ) and the th DRM field point, ( 1 ,  2 ), as depicted in Figure 2. In order to solve the unknown coefficient   , (52) should be collocated on the  DRM field points as for  = 1, 2, . . .,  and   is the distance between the th and th DRM field points.Then, the corresponding particular solution can be approximated by with the particular solutions   () given by ( 41) and (42).Also, the particular solutions of shear force and bending or twisting moment can be approximated, respectively, as In ( 55) and ( 56), the particular solutions   and Υ  are given in ( 46) and (47), respectively.
After the particular solutions are approximated by the DRM, the modified boundary conditions (51) become well defined.Therefore, it is ready to use the MFS for solving the homogeneous solution.Formally, the homogeneous solution can be approximated by where   * ( 1 ,  2 ;  1 ,  2 ) are the fundamental solutions defined by and ( 1 ,  2 ) are  source points outside the plate domain as described in Figure 3.In (57), the 3 unknowns    can be solved by collocating the modified boundary conditions (51) on  boundary points.More details of the MFS and the explicit fundamental solutions can be found in [24,26].After both the particular and homogeneous solutions are solved, the principle of superposition (48) can be applied for obtaining the desired solution.

Numerical Results
In order to validate the derived particular solutions, three numerical cases are considered.Typically,  =  = 2 and ] = 0.3 are selected in these numerical studies.Furthermore, the MFS is always arranged to be sufficient and accurate such that the numerical errors are majorly controlled by the DRM.The root mean square errors (RMSEs) of the numerical solutions are defined as where  NUM , is the numerical solution obtained by the MFS-DRM at the th point of the  considered positions and  ANA , is the corresponding analytical solution of the problem.Here,  will be chosen to be sufficiently large to ensure the utilization of the RMSEs.
In our numerical experiments, we consider the following Reissner/Mindlin plate problem: This problem has been considered for both the Mindlin [24] and the Reissner [26] cases and their analytical particular solutions can be unified as Mathematical Problems in Engineering Example 1. First, we consider a Reissner plate defined in [−1, 1] 2 subjected to clamped boundary conditions, which are set according to the analytical solutions in (61).
The particular solutions of the present problem are solved by the DRM based on the APS [26], Chebyshev polynomials [25], and PMQ.Table 1 gives the RMSEs for the solutions obtained by the DRMs with different bases and .Basically, the solutions obtained by the Chebyshev polynomials are the most accurate and the application is straightforward if the arbitrary loadings are given by functions.However, when the loadings are measured on some scattered points, a twostage method [31] is required which is relatively more difficult compared to the direct application of either APS or PMQ.In the table, the superior accuracy of the MQ over the first few orders of the APS can be observed.In addition, the accuracy improvement for the numerical solutions obtained by the high-order PMQ over the traditional MQ can also be observed.Overall, the best accuracy of the high-order PMQ is close to that of the high-order APS while the former RBF does not need augmented polynomials.In practice, it can be cumbersome to implement the particular solutions of the high-order augmented polynomials and thus the superiority of the PMQ over the APS becomes more relevant.
For both MQ and PMQ, the optimal shape parameters  are searched as demonstrated in Figure 4. Basically, the numerical accuracy improves with increasing  value.However, as  is continuously made larger, the interpolation matrix becomes ill-conditioned [32].Then, a typical error contour is given in Figure 5 for the solution obtained by the MFS-DRM based on the MQ.In the figure, the larger errors are on the boundary as expected.
Finally, the RMSE comparison for different numbers of nodes is given in Figure 6, in which higher resolution gives better accuracy.Overall, these results should have validated the polyharmonic multiquadric particular solutions of   ().Example 2.Then, we consider a similar Mindlin plate problem by changing the boundary conditions to the free boundary condition on one side.Table 2 addresses the RMSEs for the numerical results in this example.The results are very similar except that the errors are slightly larger due to the numerical approximations on the free boundary condition.
In addition, the optimal shape parameter searches and the error comparison for different numbers of nodes are given in Figures 7 and 8, respectively.The results are similar to the previous example.Therefore, the polyharmonic multiquadric particular solutions of   () and Υ  () are also validated.

Discussions
In this paper, analytical particular solutions of the polyharmonic multiquadrics associated with Reissner and Mindlin thick plates in bending are derived.In the derivations, the coupled system of the three second-order governing equations is transformed into a single sixth-order product operator of biharmonic and Helmholtz operators.Then, a method is introduced to eliminate the Helmholtz operator, which enables the use of the polyharmonic multiquadric.Numerical results are performed to validate the derived particular solutions and to demonstrate the superiority of the polyharmonic multiquadrics over the traditional multiquadrics and the augmented polynomial splines.

Figure 1 :
Figure 1: Configuration of a thick plate.

2 Figure 2 :
Figure 2: Field points of the DRM.

Figure 3 :
Figure 3: Boundary and source points of the MFS.

Figure 6 :Figure 7 :
Figure 6: RMSE comparisons of Example 1 for different numbers of nodes.

Figure 8 :
Figure 8: RMSE comparisons of Example 2 for different numbers of nodes.