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A gradient reproducing kernel collocation method for high order differential equations

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Abstract

The High order Gradient Reproducing Kernel in conjunction with the Collocation Method (HGRKCM) is introduced for solutions of 2nd- and 4th-order PDEs. All the derivative approximations appearing in PDEs are constructed using the gradient reproducing kernels. Consequently, the computational cost for construction of derivative approximations reduces tremendously, basis functions for derivative approximations are smooth, and the accumulated error arising from calculating derivative approximations are controlled in comparison to the direct derivative counterparts. Furthermore, it is theoretically estimated and numerically tested that the same number of collocation points as the source points can be used to obtain the optimal solution in the HGRKCM. Overall, the HGRKCM is roughly 10–25 times faster than the conventional reproducing kernel collocation method. The convergence of the present method is estimated using the least squares functional equivalence. Numerical results are verified and compared with other strong-form-based and Galerkin-based methods.

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Acknowledgements

Research reported in this paper was partially supported by DoD SERDP under contract number W912HQ18C0099.

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Authors and Affiliations

  1. Department of Civil and Material Engineering, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Ashkan Mahdavi & Sheng-Wei Chi

  2. Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, 39406, USA

    Huiqing Zhu

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  1. Ashkan Mahdavi
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  2. Sheng-Wei Chi
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  3. Huiqing Zhu
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Correspondence to Sheng-Wei Chi.

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Appendices

Appendix A

To demonstrate how exactly the procedure shown in Sect. 2.4 works for constructing gradient reproducing kernel shape function’s derivative of a desired order, second derivative approximation of the two-dimensional vector field \(\varvec{u}\) is considered:

$$\begin{aligned}&\varvec{u}_{,xx} \approx \varvec{w}_{xx}=\sum ^{N_s}_{I=1}\Psi ^{xx}_I(\varvec{x})\varvec{d}_I \end{aligned}$$
(A.1a)
$$\begin{aligned}&\varvec{u}_{,yy} \approx \varvec{w}_{yy}=\sum ^{N_s}_{I=1}\Psi ^{yy}_I(\varvec{x})\varvec{d}_I \end{aligned}$$
(A.1b)
$$\begin{aligned}&\varvec{u}_{,xy} \approx \varvec{w}_{xy}=\sum ^{N_s}_{I=1}\Psi ^{xy}_I(\varvec{x})\varvec{d}_I \end{aligned}$$
(A.1c)

where \(\varvec{w}_{xx}\), \(\varvec{w}_{xx}\), and \(\varvec{w}_{xy}\) are the gradient second order derivatives approximation of vector field \(\varvec{u}\). Recalling (19),

$$\begin{aligned}&\Psi ^{xx}_I = \varvec{C}^1(\varvec{x};\varvec{x}-\varvec{x}_I) \phi _a(\varvec{x}-\varvec{x}_I) \end{aligned}$$
(A.2a)
$$\begin{aligned}&\Psi ^{yy}_I = \varvec{C}^2(\varvec{x};\varvec{x}-\varvec{x}_I) \phi _a(\varvec{x}-\varvec{x}_I) \end{aligned}$$
(A.2b)
$$\begin{aligned}&\Psi ^{xy}_I = \varvec{C}^3(\varvec{x};\varvec{x}-\varvec{x}_I) \phi _a(\varvec{x}-\varvec{x}_I) \end{aligned}$$
(A.2c)

where \(\varvec{C}^i\), \(i=1, 2, 3\), are the corresponding correction functions for each case constructed by the related coefficients vector and quadratic basis vector,

$$\begin{aligned} \varvec{C}^i(\varvec{x};\varvec{x}-\varvec{x}_I) = \varvec{H}^T(\varvec{x}-\varvec{x}_I)\varvec{b}^i(\varvec{x}), \quad i=1,2,3 \end{aligned}$$
(A.3)

where \(\varvec{b}^i\) in each case is obtained by satisfying the partition of nullity and derivative reproducing conditions, in this case second order derivatives reproducing conditions.

For the second derivative with respect to x the reproducing condition are,

$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})=0 \end{aligned}$$
(A.4a)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})x_I=0 \end{aligned}$$
(A.4b)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})y_I=0 \end{aligned}$$
(A.4c)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})x^2_I=2 \end{aligned}$$
(A.4d)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})y^2_I=0 \end{aligned}$$
(A.4e)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi ^{xx}(\varvec{x})x_I y_I(\varvec{x})=0 \end{aligned}$$
(A.4f)

By multiplying (A.4a) by x and subtracting (A.4b):

$$\begin{aligned} \sum ^{N_s}_{I=1} \Psi ^{xx}_I(\varvec{x})(x-x_I) = 0 \end{aligned}$$
(A.5)

Multiplying (A.4a) by y and subtracting (A.4c) results in:

$$\begin{aligned} \sum ^{N_s}_{I=1} \Psi ^{xx}_I(\varvec{x})(y-y_I) = 0 \end{aligned}$$
(A.6)

For quadratic terms, By multiplying (A.4a) by \(x^2\), adding (A.4d), and then subtracting two times of (A.4f),

$$\begin{aligned}&\sum ^{N_s}_{I=1} \Psi ^{xx}_I(\varvec{x})(x^2 + x_I^2 - 2 x x_I)\nonumber \\&\quad =\sum ^{N_s}_{I=1} \Psi ^{xx}_I (x-x_I)^2=2 \end{aligned}$$
(A.7)

By following the same procedure and multiplying (A.4a) by \(y^2\) and adding (A.4e) and abstracting two time of (A.4f),

$$\begin{aligned} \sum ^{N_s}_{I=1} \Psi ^{xx}_I (y-y_I)^2=0 \end{aligned}$$
(A.8)

and finally, for the last term, by multiplying (A.4a) by xy, \(-y\) by (A.4b), \(-x\) by (A.4c), and summing up all these expressions with (A.4f),

$$\begin{aligned}&\sum ^{N_s}_{I=1} \Psi ^{xx}_I(xy-y x_I-x y_I+x_I y_I)\nonumber \\&\quad =\sum ^{N_s}_{I=1} \Psi ^{xx}_I(x-x_I)(y-y_I)=0 \end{aligned}$$
(A.9)

Writing all Eqs. (A.5)–(A.9) and including the partition of nullity in (A.4a),

$$\begin{aligned} \sum ^{N_s}_{I=1}\Psi _I^{xx}(\varvec{x}) \underbrace{ \left[ \begin{array}{c} 1 \\ x-x_I \\ y-y_I\\ (x-x_I)^2\\ (y-y_I)^2\\ (x-x_I)(y-y_I) \end{array} \right] }_{\varvec{H}(\varvec{x}-\varvec{x}_I)} = \underbrace{ \left[ \begin{array}{c} 0\\ 0\\ 0\\ 2\\ 0\\ 0 \end{array} \right] }_{\varvec{H}_{,xx}(\varvec{0})} \end{aligned}$$
(A.10)

The right-hand side in (A.10) is the second-order explicit derivative of quadratic basis vector where \(\varvec{x}-\varvec{x}_I=0\). By taking the same steps for satisfying \(\Psi ^{yy}\) and \(\Psi ^{xy}\) all the second-order derivative reproducing conditions become,

$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi _I^{xx}(\varvec{x}) \varvec{H}(\varvec{x}-\varvec{x}_I) = \varvec{H}_{,xx}(\varvec{0})\end{aligned}$$
(A.11a)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi _I^{yy}(\varvec{x}) \varvec{H}(\varvec{x}-\varvec{x}_I) = \varvec{H}_{,yy}(\varvec{0})\end{aligned}$$
(A.11b)
$$\begin{aligned}&\sum ^{N_s}_{I=1}\Psi _I^{xy}(\varvec{x}) \varvec{H}(\varvec{x}-\varvec{x}_I) = \varvec{H}_{,xy}(\varvec{0}) \end{aligned}$$
(A.11c)

By using the definitions of gradient derivatives of RK function in (A.2) and correction function in (A.3) and substituting those into Eq. (A.11),

$$\begin{aligned}&\varvec{M}(\varvec{x})\varvec{b}^1(\varvec{x})=\varvec{H}_{,xx}(\varvec{0}) \end{aligned}$$
(A.12a)
$$\begin{aligned}&\varvec{M}(\varvec{x})\varvec{b}^2(\varvec{x})=\varvec{H}_{,yy}(\varvec{0})\end{aligned}$$
(A.12b)
$$\begin{aligned}&\varvec{M}(\varvec{x})\varvec{b}^3(\varvec{x})=\varvec{H}_{,xy}(\varvec{0}) \end{aligned}$$
(A.12c)

where \(\varvec{M}\) is the moment matrix defined in (8). Eventually, by having coefficients vector in (A.12), the second-order gradient derivatives are derived as,

$$\begin{aligned}&\Psi ^{xx}_{I}(\varvec{x})=\varvec{H}^{T}_{,xx}(\varvec{0})~\varvec{M}^{-1}(\varvec{x})~\varvec{H}(\varvec{x}-\varvec{x}_I)~\phi _a(\varvec{x}-\varvec{x}_I) \end{aligned}$$
(A.13a)
$$\begin{aligned}&\Psi ^{yy}_{I}(\varvec{x})=\varvec{H}^{T}_{,yy}(\varvec{0})~\varvec{M}^{-1}(\varvec{x})~\varvec{H}(\varvec{x}-\varvec{x}_I)~\phi _a(\varvec{x}-\varvec{x}_I)\end{aligned}$$
(A.13b)
$$\begin{aligned}&\Psi ^{xy}_{I}(\varvec{x})=\varvec{H}^{T}_{,xy}(\varvec{0})~\varvec{M}^{-1}(\varvec{x})~\varvec{H}(\varvec{x}-\varvec{x}_I)~\phi _a(\varvec{x}-\varvec{x}_I) \end{aligned}$$
(A.13c)

Obviously, equations in (A.13) are just different by their first terms, which indicates that the method is straightforward and computationally efficient. More importantly, the inversion of moment matrix derivatives, which suffer from ill-conditioning and large condition numbers are no longer needed to be calculated. Following the same rule, higher order gradient derivatives could be obtained.

Appendix B

Sub-matrices shown in Eq. (54) are shown in the following. \(\varvec{A}^i\) (\(i=1,\ldots ,13\)) in the case of 4th-order Kirchhoff plate discrete equation is defined as,

$$\begin{aligned} \varvec{A}^1= & {} \begin{pmatrix} {L}^1 \varvec{\Psi }_{xxxx}^T({x}^d_1) \\ {L}^1 \varvec{\Psi }_{xxxx}^T({x}^d_2)\\ \vdots \\ {L}^1 \varvec{\Psi }_{xxxx}^T({x}^d_{N_d}) \end{pmatrix},~ \varvec{A}^2= \begin{pmatrix} {L}^2 \varvec{\Psi }_{xxyy}^T({x}^d_1) \\ {L}^2 \varvec{\Psi }_{xxyy}^T({x}^d_2)\\ \vdots \\ {L}^2 \varvec{\Psi }_{xxyy}^T({x}^d_{N_d}) \end{pmatrix},\nonumber \\ \varvec{A}^3= & {} \begin{pmatrix} {L}^3 \varvec{\Psi }_{yyyy}^T({x}^d_1) \\ {L}^3 \varvec{\Psi }_{yyyy}^T({x}^d_2)\\ \vdots \\ {L}^3 \varvec{\Psi }_{yyyy}^T({x}^d_{N_d}) \end{pmatrix},~ \varvec{A}^4=\begin{pmatrix} {B}_g \varvec{\Psi }^{T}({x}^g_1) \\ {B}_g \varvec{\Psi }^{T}({x}^g_2)\\ \vdots \\ {B}_g \varvec{\Psi }^{T}({x}^g_{N_g}) \end{pmatrix},\nonumber \\ \varvec{A}^5= & {} \begin{pmatrix} {B}_h^1 \varvec{\Psi }_{x}^T({x}^h_1) \\ {B}_h^1 \varvec{\Psi }_{x}^T({x}^h_2)\\ \vdots \\ {B}_h^1 \varvec{\Psi }_{x}^T({x}^h_{N_h}) \end{pmatrix},~ \varvec{A}^6=\begin{pmatrix} {B}_h^2 \varvec{\Psi }_{y}^T({x}^h_1) \\ {B}_h^2 \varvec{\Psi }_{y}^T({x}^h_2)\\ \vdots \\ {B}_h^2 \varvec{\Psi }_{y}^T({x}^h_{N_h}) \end{pmatrix},\nonumber \\ \varvec{A}^7= & {} \begin{pmatrix} B_m^1 \varvec{\Psi }_{xx}^T({x}^m_1) \\ B_m^1 \varvec{\Psi }_{xx}^T({x}^m_2)\\ \vdots \\ B_m^1 \varvec{\Psi }_{xx}^T({x}^m_{N_m}) \end{pmatrix},~ \varvec{A}^8= \begin{pmatrix} B_m^2 \varvec{\Psi }_{yy}^T({x}^m_1) \\ B_m^2 \varvec{\Psi }_{yy}^T({x}^m_2)\\ \vdots \\ B_m^2 \varvec{\Psi }_{yy}^T({x}^m_{N_m}) \end{pmatrix},\nonumber \\ \varvec{A}^9= & {} \begin{pmatrix} B_m^3 \varvec{\Psi }_{xy}^T({x}^m_1) \\ B_m^3 \varvec{\Psi }_{xy}^T({x}^m_2)\\ \vdots \\ B_m^3 \varvec{\Psi }_{xy}^T({x}^m_{N_m}) \end{pmatrix},~ \varvec{A}^{10}= \begin{pmatrix} B_m^1 \varvec{\Psi }_{xxx}^T({x}^q_1) \\ B_m^1 \varvec{\Psi }_{xxx}^T({x}^q_2)\\ \vdots \\ B_m^1 \varvec{\Psi }_{xxx}^T({x}^q_{N_q}) \end{pmatrix},\nonumber \\ \varvec{A}^{11}= & {} \begin{pmatrix} B_m^2 \varvec{\Psi }_{yyy}^T({x}^q_1) \\ B_m^2 \varvec{\Psi }_{yyy}^T({x}^q_2)\\ \vdots \\ B_m^2 \varvec{\Psi }_{yyy}^T({x}^q_{N_q}) \end{pmatrix},~ \varvec{A}^{12}= \begin{pmatrix} B_m^3 \varvec{\Psi }_{xxy}^T({x}^q_1) \\ B_m^3 \varvec{\Psi }_{xxy}^T({x}^q_2)\\ \vdots \\ B_m^3 \varvec{\Psi }_{xxy}^T({x}^q_{N_q}) \end{pmatrix},\nonumber \\ \varvec{A}^{13}= & {} \begin{pmatrix} B_m^4 \varvec{\Psi }_{xyy}^T({x}^q_1) \\ B_m^4 \varvec{\Psi }_{xyy}^T({x}^q_2)\\ \vdots \\ B_m^4 \varvec{\Psi }_{xyy}^T({x}^q_{N_q}) \end{pmatrix}, \end{aligned}$$
(B.1)

and for sub-matrices \(\varvec{b}^i\) (\(i=1,\ldots ,5\)),

$$\begin{aligned} \varvec{b}^1= & {} \begin{pmatrix} \varvec{f}({x}^d_1) \\ \varvec{f}({x}^d_2) \\ \vdots \\ \varvec{f}({x}^d_{N_d}) \end{pmatrix} ,\varvec{b}^2= \begin{pmatrix} \varvec{u}_g({x}^g_1) \\ \varvec{u}_g({x}^g_2) \\ \vdots \\ \varvec{u}_g({x}^g_{N_g}) \end{pmatrix} ,\varvec{b}^3= \begin{pmatrix} \varvec{h}({x}^h_1) \\ \varvec{h}({x}^h_2) \\ \vdots \\ \varvec{h}({x}^h_{N_h}) \end{pmatrix}\nonumber \\ ,\varvec{b}^4= & {} \begin{pmatrix} \varvec{M}({x}^m_1) \\ \varvec{M}({x}^m_2) \\ \vdots \\ \varvec{M}({x}^m_{N_m}) \end{pmatrix} ,\varvec{b}^5= \begin{pmatrix} \varvec{Q}({x}^q_1) \\ \varvec{Q}({x}^q_2) \\ \vdots \\ \varvec{Q}({x}^q_{N_q}) \end{pmatrix} \end{aligned}$$
(B.2)

where \(N_d\) denotes the domain collocation points. \(N_g\), \(N_h\), \(N_m\), and \(N_q\) are the numbers of displacement, rotation, moment and shear boundary collocation points, respectively.

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Mahdavi, A., Chi, SW. & Zhu, H. A gradient reproducing kernel collocation method for high order differential equations. Comput Mech 64, 1421–1454 (2019). https://doi.org/10.1007/s00466-019-01724-0

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