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Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates

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Abstract

We describe novel numerical methods for solving a class of high-order time-dependent PDEs on general geometries, which involve second-order derivatives in time and up-to fourth-order derivatives in space. This type of PDEs are widely used in applications such as the Boussinesq equation and in modeling thin-walled elastic structures such as beams, plates and shells, etc. High-order spatial derivatives together with general geometries bring a number of challenges for many numerical methods. In this paper, we resolve these challenges by discretizing the spatial derivatives in domains with general geometries using the composite overlapping grid method. The discretization on overlapping grids requires numerical interpolations to couple solutions on different component grids. However, the presence of interpolation equations breaks the symmetry of the overall spatial discretization, causing numerical instability in time-stepping schemes. To address this, a fourth-order hyper-dissipation term is included for stabilization. Investigation of incorporating the hyper-dissipation term into several time-stepping schemes for solving the semi-discrete system leads to the development of a series of algorithms. Accurate and stable numerical boundary conditions for Dirichlet and Neumann type boundaries are also developed for general geometries. Quadratic eigenvalue problems for a simplified model problem on 1D overlapping grids are considered to reveal the weak instability caused by interpolation between component grids. This model problem is also investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments and two benchmark problems concerning the Kirchhoff–Love plate model are presented to demonstrate the accuracy and efficiency of our approaches. This work shows that finite difference methods on overlapping grids are well-suited for solving high-order PDEs in complex domains for realistic applications.

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Notes

  1. Overture is freely available from http://www.overtureframework.org.

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Acknowledgements

L. Li is grateful to Professor W.D. Henshaw of Rensselaer Polytechnic Institute (RPI) for helpful conversations.

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  1. The MathWorks, Inc., Natick, MA, 01760, USA

    Longfei Li

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Hangjie Ji

  3. Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Qi Tang

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L. Li: Research supported by the Louisiana Board of Regents Support Fund under contract No. LEQSF(2018-21)-RD-A-23.

Q. Tang: Research supported by the U.S. Department of Energy (DOE), Office of Advanced Scientific Computing Research (ASCR) through DOE-FOA-2493.

Appendix A

Appendix A

1.1 Appendix A.1: Transformed PDE on reference domain

The specific formula for the equation (1) transformed into the reference domain is given by

$$\begin{aligned} M\frac{\partial ^2 W}{\partial t^2}={{\mathcal {L}}}W + F, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {L}} W&=\left( -K\right) \,W\\&\quad +\left( T\,\left( r_{xx}+r_{yy}\right) -D\,\left( r_{xxxx}+2\,r_{xxyy}+r_{yyyy}\right) \right) \,{W_{r}}\\&\quad +\left( T\,\left( s_{xx}+s_{yy}\right) -D\,\left( s_{xxxx}+2\,s_{xxyy}+s_{yyyy}\right) \right) \,{W_{s}}\\&\quad +\left( T\,\left( r_{x}^2+r_{y}^2\right) -D\,\left( 3\,r_{xx}^2+2\,r_{xx}\,r_{yy}+4\,r_{xy}^2+3\,r_{yy}^2\right. \right. \\&\left. \left. \quad +4\,r_{x}\,r_{xxx}+4\,r_{x}\,r_{xyy}+4\,r_{xxy}\,r_{y}+4\,r_{y}\,r_{yyy}\right) \right) \,{W_{rr}}\\&\quad +(T\,\left( 2\,r_{x}\,s_{x}+2\,r_{y}\,s_{y}\right) -D\,(4\,r_{x}\,s_{xxx}+4\,r_{xxx}\,s_{x}+4\,r_{x}\,s_{xyy}+4\,r_{xyy}\,s_{x}+6\,r_{xx}\,s_{xx}\\&\quad +8\,r_{xy}\,s_{xy}+4\,r_{xxy}\,s_{y}+4\,r_{y}\,s_{xxy}+2\,r_{xx}\,s_{yy}+2\,r_{yy}\,s_{xx}\\&\quad +4\,r_{y}\,s_{yyy}+4\,r_{yyy}\,s_{y}+6\,r_{yy}\,s_{yy}))\,{W_{rs}}\\&\quad +\left( T\,\left( s_{x}^2+s_{y}^2\right) -D\,\left( 3\,s_{xx}^2+2\,s_{xx}\,s_{yy}\right. \right. \\&\left. \left. \quad +4\,s_{xy}^2+3\,s_{yy}^2+4\,s_{x}\,s_{xxx}+4\,s_{x}\,s_{xyy}+4\,s_{xxy}\,s_{y}+4\,s_{y}\,s_{yyy}\right) \right) \,{W_{ss}}\\&\quad +\left( -D\,\left( 6\,r_{x}^2\,r_{xx}+2\,r_{xx}\,r_{y}^2+6\,r_{y}^2\,r_{yy}+2\,r_{x}\,\left( r_{x}\,r_{yy}+2\,r_{xy}\,r_{y}\right) +4\,r_{x}\,r_{xy}\,r_{y}\right) \right) \,{W_{rrr}}\\&\quad +(-D\,(2\,r_{x}\,\left( r_{x}\,s_{yy}+2\,r_{xy}\,s_{y}+2\,r_{y}\,s_{xy}+r_{yy}\,s_{x}\right) +3\,r_{x}^2\,s_{xx}+2\,r_{y}^2\,s_{xx}+3\,r_{y}^2\,s_{yy}\\&\quad +2\,s_{x}\,\left( r_{x}\,r_{yy}+2\,r_{xy}\,r_{y}\right) +r_{x}\,\left( 3\,r_{x}\,s_{xx}+3\,r_{xx}\,s_{x}\right) \\&\quad +4\,r_{xy}\,\left( r_{x}\,s_{y}+r_{y}\,s_{x}\right) +r_{y}\,\left( 3\,r_{y}\,s_{yy}+3\,r_{yy}\,s_{y}\right) \\&\quad +9\,r_{x}\,r_{xx}\,s_{x}+4\,r_{x}\,r_{y}\,s_{xy}+4\,r_{xx}\,r_{y}\,s_{y}+9\,r_{y}\,r_{yy}\,s_{y}))\,{W_{rrs}}\\&\quad +(-D\,(2\,s_{x}\,\left( r_{x}\,s_{yy}+2\,r_{xy}\,s_{y}+2\,r_{y}\,s_{xy}+r_{yy}\,s_{x}\right) +3\,r_{xx}\,s_{x}^2+2\,r_{xx}\,s_{y}^2+3\,r_{yy}\,s_{y}^2\\&\quad +s_{x}\,\left( 3\,r_{x}\,s_{xx}+3\,r_{xx}\,s_{x}\right) +4\,s_{xy}\,\left( r_{x}\,s_{y}+r_{y}\,s_{x}\right) \\&\quad +s_{y}\,\left( 3\,r_{y}\,s_{yy}+3\,r_{yy}\,s_{y}\right) +2\,r_{x}\,\left( s_{x}\,s_{yy}+2\,s_{xy}\,s_{y}\right) \\&\quad +9\,r_{x}\,s_{x}\,s_{xx}+4\,r_{xy}\,s_{x}\,s_{y}+4\,r_{y}\,s_{xx}\,s_{y}+9\,r_{y}\,s_{y}\,s_{yy}))\,{W_{rss}}\\&\quad +\left( -D\,\left( 6\,s_{x}^2\,s_{xx}+2\,s_{xx}\,s_{y}^2+6\,s_{y}^2\,s_{yy}+2\,s_{x}\,\left( s_{x}\,s_{yy}+2\,s_{xy}\,s_{y}\right) +4\,s_{x}\,s_{xy}\,s_{y}\right) \right) \,{W_{sss}}\\&\quad +\left( -D\,\left( r_{x}^4+2\,r_{x}^2\,r_{y}^2+r_{y}^4\right) \right) \,{W_{rrrr}}\\&\quad +\left( -D\,\left( r_{x}\,\left( s_{x}\,r_{y}^2+2\,r_{x}\,s_{y}\,r_{y}\right) \,2+4\,r_{x}^3\,s_{x}+4\,r_{y}^3\,s_{y}+2\,r_{x}\,r_{y}^2\,s_{x}\right) \right) \,{W_{rrrs}}\\&\quad +\left( -D\,\left( 6\,r_{x}^2\,s_{x}^2+6\,r_{y}^2\,s_{y}^2+r_{x}\,\left( r_{x}\,s_{y}^2+2\,r_{y}\,s_{x}\,s_{y}\right) \,2+s_{x}\,\left( s_{x}\,r_{y}^2+2\,r_{x}\,s_{y}\,r_{y}\right) \,2\right) \right) \,{W_{rrss}}\\&\quad +\left( -D\,\left( s_{x}\,\left( r_{x}\,s_{y}^2+2\,r_{y}\,s_{x}\,s_{y}\right) \,2+4\,r_{x}\,s_{x}^3+4\,r_{y}\,s_{y}^3+2\,r_{x}\,s_{x}\,s_{y}^2\right) \right) \,{W_{rsss}}\\&\quad +\left( -D\,\left( s_{x}^4+2\,s_{x}^2\,s_{y}^2+s_{y}^4\right) \right) \,{W_{ssss}}. \end{aligned}$$

The definitions of coefficients \(a_i({\textbf{r}}), b_{ij}({\textbf{r}}),c_{ijk}({\textbf{r}}),d_{ijkl}({\textbf{r}})\) (coefficients in front of the red terms) can be readily read off the above formula for \({{\mathcal {L}}}W\).

1.2 Appendix A.2: Formula of the Discrete transformation

The Fourier transform (symbol) of the discrete operator \({{\mathcal {L}}}_h\) is

$$\begin{aligned} {\hat{Q}}(\xi _r,\xi _s;{\textbf{r}})&=-K-b_{11}\frac{4\sin ^2(\xi _r/2)}{h_r^2}-b_{12} \frac{\sin (\xi _r)\sin (\xi _s)}{h_r h_s}\\&\quad -b_{22}\frac{4\sin ^2(\xi _s/2)}{h_s^2}+d_{1111}\frac{16\sin ^4(\xi _r/2)}{h_r^4}\\&\quad +d_{1112}\frac{4\sin ^2(\xi _r/2)\sin (\xi _r)\sin (\xi _s)}{h_r^3h_s}\\&\quad +d_{1122}\frac{16\sin ^2(\xi _r/2)\sin ^2(\xi _s/2)}{h_r^2h_s^2}+d_{1222}\frac{4\sin (\xi _r)\sin ^2(\xi _s/2)\sin (\xi _s)}{h_rh_s^3}\\&\quad +d_{2222}\frac{16\sin ^4(\xi _s/2)}{h_s^4} \end{aligned}$$

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Li, L., Ji, H. & Tang, Q. Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates. J Sci Comput 98, 41 (2024). https://doi.org/10.1007/s10915-023-02430-z

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