RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations

Abstract

We present an open-source software RKPM2D for solving PDEs under the reproducing kernel particle method (RKPM)-based meshfree computational framework. Compared to conventional mesh-based methods, RKPM provides many attractive features, such as arbitrary order of continuity and discontinuity, relaxed tie between the quality of the discretization and the quality of approximation, simple h-adaptive refinement, and ability to embed physics-based enrichment functions, among others, which make RKPM promising for solving challenging engineering problems. The aim of the present software package is to support reproducible research and serve as an efficient test platform for further development of meshfree methods. The RKPM2D software consists of a set of data structures and subroutines for discretizing two-dimensional domains, nodal representative domain creation by Voronoi diagram partitioning, boundary condition specification, reproducing kernel shape function generation, domain integrations with stabilization, a complete meshfree solver, and visualization tools for post-processing. In this paper, a brief overview that covers the key theoretical aspects of RKPM is given, such as the reproducing kernel approximation, weak form using Nitsche’s method for boundary condition enforcement, various domain integration schemes (Gauss quadrature and stabilized nodal integration methods), as well as the fully discrete equations. In addition, the computer implementation aspects employed in RKPM2D are discussed in detail. Benchmark problems solved by RKPM2D are presented to demonstrate the convergence, efficiency, and robustness of the RKPM implementation.

Appendices

Appendix A

In RKPM2D, six kernel functions with different levels of continuity are implemented, as described in Table 1, and the corresponding mathematical expressions of these kernel functions are given as follows.

• 1. The Heaviside kernel function:

  $$ \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} {\text{for}} \\ {\text{for}} \\ \end{array} } & {\begin{array}{*{20}c} {0 \le z_{I} \le 1,} \\ {z_{I} > 1} \\ \end{array} } \\ \end{array} } \right. $$

  (68)

• 2. The linear B-spline (tent) kernel function:

  $$ \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \left\{ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {1 - z_{I} } \\ 0 \\ \end{array} } & {\begin{array}{*{20}l} {\text{for}} \\ {\text{for}} \\ \end{array} } & {\begin{array}{*{20}l} {0 \le z_{I} \le 1,} \\ {z_{I} > 1} \\ \end{array} } \\ \end{array} } \right. $$

  (69)

• 3. The quadratic B-spline kernel function:

  $$ \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \left\{ {\begin{array}{*{20}l} {1 - 3z_{I}^{2} } \\ {3/2 - 3z_{I} + 3/2z_{I}^{2} } \\ 0 \\ \end{array} } \right.\begin{array}{*{20}l} {\text{for}} \\ {\text{for}} \\ {\text{for}} \\ \end{array} \begin{array}{*{20}l} {0 \le z_{I} \le 1/3,} \\ {1/3 \le z_{I} \le 1,} \\ {z_{I} > 1} \\ \end{array} $$

  (70)

• 4. The cubic B-spline kernel function:

  $$ \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \left\{ {\begin{array}{*{20}l} {2/3 - 4z_{I}^{2} + 4z_{I}^{3} } \\ {4/3 - 4z_{I} + 4z_{I}^{2} - 4/3z_{I}^{3} } \\ 0 \\ \end{array} } \right.\begin{array}{*{20}l} \quad{\text{for}} \\ \quad {\text{for}} \\ \quad{\text{for}} \\ \end{array} \begin{array}{*{20}l} \quad {0 \le z_{I} \le 1/2,} \\ \quad {1/2 \le z_{I} \le 1,} \\ \quad{z_{I} > 1} \\ \end{array} $$

  (71)

• 5. The quartic B-spline kernel function:

  $$ \begin{aligned} & \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \\ & \left\{ {\begin{array}{*{20}l} {1 - \frac{150}{23}z_{I}^{2} + \frac{375}{23}z_{I}^{4} } \hfill & { {\text{for}} 0 \le z_{I} \le \frac{1}{5},} \hfill \\ {\frac{22}{23} + \frac{20}{23}z_{I} - \frac{300}{23}z_{I}^{2} + \frac{500}{23}z_{I}^{3} - \frac{250}{23}z_{I}^{4} } \hfill & {{\text{for}} \frac{1}{5} \le z_{I} \le \frac{3}{5},} \hfill \\ {\frac{125}{46} - \frac{250}{23} z_{I} + \frac{375}{23}z_{I}^{2} - \frac{250}{23}z_{I}^{3} + \frac{125}{46}z_{I}^{4} } \hfill & {{\text{for}} \frac{3}{5} \le z_{I} \le 1,} \hfill \\ 0 \hfill & { {\text{for}} z_{I} > 1} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

  (72)

• 6. The quintic B-spline kernel function:

  $$ \begin{aligned} & \varPhi_{a} \left( {\varvec{x} - \varvec{x}_{I} } \right) = \\ & \left\{ {\begin{array}{*{20}l} {1 - \frac{90}{11}z_{I}^{2} + \frac{405}{11}z_{I}^{4} - \frac{405}{11}z_{I}^{5} } \hfill & {{\text{for}}\,\, 0 \le z_{I} \le \frac{1}{3},} \hfill \\ {\frac{17}{22} + \frac{75}{22} z_{I} - \frac{315}{11}z_{I}^{2} + \frac{674}{11}z_{I}^{3} - \frac{1215}{22}z_{I}^{4} + \frac{405}{22}z_{I}^{5} } \hfill & {{\text{for}}\,\, \frac{1}{3} \le z_{I} \le \frac{2}{3},} \hfill \\ {\frac{81}{22} - \frac{405}{22} z_{I} + \frac{405}{11}z_{I}^{2} - \frac{405}{11}z_{I}^{3} + \frac{405}{22}z_{I}^{4} - \frac{81}{22}z_{I}^{5} } \hfill & {{\text{for}}\,\, \frac{2}{3} \le z_{I} \le 1,} \hfill \\ 0 \hfill & {{\text{for}}\,\, z_{I} > 1} \hfill \\ \end{array} } \right. \\ \end{aligned} $$

  (73)

Appendix B

In this section, a diffusion problem is used as example to illustrate how to modify RKPM2D for the solution of different types of PDEs. A diffusion equation is considered as follows:

$$ \begin{array}{*{20}l} {\left( {D_{ij} u_{,j} } \right)_{,i} + b = 0} \hfill & {{\text{on}}\,\varvec{ }\varOmega } \hfill \\ {D_{ij} u_{,j} n_{i} = t} \hfill & {{\text{on}}\,\partial \varOmega_{t} } \hfill \\ {u = g} \hfill & {{\text{on}}\,\partial \varOmega_{g} } \hfill \\ \end{array} $$

(74)

where $u$ is a scalar field, $D_{ij}$ is the diffusivity, $b$ is the source term, and $t$ and $g$ are the prescribed boundary flux and boundary values of $u$ on $\partial \varOmega_{t}$ and $\partial \varOmega_{g}$, respectively. By introducing the RK approximation in Eq. (11), (74) can be recast into the following matrix equations for isotropic scalar diffusivity:

$$ \mathop \sum \limits_{J} K_{IJ} u_{J} - F_{I} = 0 $$

(75)

where

$$ K_{IJ} = K_{IJ}^{d} + K_{IJ}^{\beta } - \left( {K_{IJ}^{g} + \mathop {K_{IJ}^{g} }\nolimits^{\text{T}} } \right) $$

(76)

$$ F_{I} = F_{I}^{\text{b}} + F_{I}^{\text{t}} + F_{IJ}^{\beta } - F_{I}^{g} $$

(77)

in which the matrices and vectors in nodal integration are expressed as

$$ K_{IJ}^{d} = \int\limits_{\varOmega } {\varvec{B}_{I}^{\text{T}} \left( \varvec{x} \right)\varvec{DB}_{J} \left( \varvec{x} \right){\text{d}}\varOmega } \approx \mathop \sum \limits_{N = 1}^{\text{NP}} \varvec{B}_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)\varvec{DB}_{J} \left( {\varvec{x}_{N} } \right)A_{N} $$

(78)

$$ F_{I}^{\text{b}} = \int\limits_{\varOmega } {\varPsi_{I}^{\text{T}} \left( \varvec{x} \right)b\left( \varvec{x} \right){\text{d}}\varOmega } \approx \mathop \sum \limits_{N = 1}^{\text{NP}} \varPsi_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)b\left( {\varvec{x}_{N} } \right)A_{N} $$

(79)

$$ F_{I}^{\text{t}} = \int\limits_{{\partial \varOmega_{t} }} {\varPsi_{I}^{\text{T}} \left( \varvec{x} \right)t\left( \varvec{x} \right){\text{d}}\varGamma } \approx \mathop \sum \limits_{N = 1}^{{{\text{NP}}t}} \varPsi_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)t\left( {\varvec{x}_{N} } \right)L_{N} $$

(80)

$$ K_{IJ}^{\beta } = \beta \int\limits_{{\partial \varOmega_{g} }} {\varPsi_{I}^{\text{T}} \left( \varvec{x} \right)S\varPsi_{J} \left( \varvec{x} \right){\text{d}}\varGamma } \approx \beta \mathop \sum \limits_{N = 1}^{{{\text{NP}}g}} \varPsi_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)S\varPsi_{J} \left( {\varvec{x}_{N} } \right)L_{N} $$

(81)

$$ K_{IJ}^{g} = \int\limits_{{\partial \varOmega_{g} }} {\varvec{B}_{I}^{\text{T}} \left( \varvec{x} \right)\varvec{D\eta }S\varPsi_{J} \left( \varvec{x} \right){\text{d}}\varGamma } \approx \mathop \sum \limits_{N = 1}^{{{\text{NP}}g}} \varvec{B}_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)\varvec{D\eta }S\varPsi_{J} \left( {\varvec{x}_{N} } \right)L_{N} $$

(82)

$$ F_{IJ}^{\beta } = \beta \int\limits_{{\partial \varOmega_{g} }} {\varPsi_{I}^{\text{T}} \left( \varvec{x} \right)Sg{\text{d}}\varGamma } \approx \beta \mathop \sum \limits_{N = 1}^{{{\text{NP}}g}} \varPsi_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)SgL_{N} $$

(83)

$$ F_{I}^{g} = \int\limits_{{\partial \varOmega_{g} }} {\varvec{B}_{I}^{\text{T}} \left( \varvec{x} \right)\varvec{D\eta }Sg{\text{d}}\varGamma } \approx \mathop \sum \limits_{N = 1}^{{{\text{NP}}g}} \varvec{B}_{I}^{\text{T}} \left( {\varvec{x}_{N} } \right)\varvec{D\eta }SgL_{N} $$

(84)

$$ \varvec{B}_{I} \left( {\varvec{x}_{N} } \right) = \left[ {\begin{array}{*{20}l} {\varPsi_{I,1} \left( {\varvec{x}_{N} } \right)} \\ {\varPsi_{I,2} \left( {\varvec{x}_{N} } \right)} \\ \end{array} } \right], \varvec{D} = \left[ {\begin{array}{*{20}l} d & 0 \\ 0 & d \\ \end{array} } \right],\varvec{ \eta } = \left[ {\begin{array}{*{20}l} {n_{1} } \\ {n_{2} } \\ \end{array} } \right],\varvec{ }S = 1. $$

(85)

where $\varvec{D}$ is the diffusivity tensor, $d$ is the diffusion coefficient, and $\varvec{\eta}$ is a collection of components of the surface unit normal on the boundary, and $S = 1$ is set for the convenience of keeping a unified coding structure in RKPM2D. Let us consider a diffusion problem [Eq. (74)] with a manufactured solution:

$$ u^{\text{exact}} = 0.1 + 0.1x_{1} + 0.2x_{2} $$

(86)

in a circular domain $\varOmega \subset {\mathbb{R}}^{2}$ shown in Fig. 9 from Sect. 3.2. The flux $t = 0.1n_{1} + 0.2n_{2}$ is imposed on $\partial \varOmega_{t} :\left( {x_{1} ,x_{2} } \right) \in \partial \varOmega , \quad x_{2} > 0.5$ where $n_{1} {\text{ and }}n_{2}$ are the normal vector components. $g = u^{\text{exact}}$ is enforced on $\partial \varOmega_{g} :\left( {x_{1} ,x_{2} } \right) \in \partial \varOmega ,\quad x_{2} \le 0.5$, and the body source is $b = 0.$

The input file for this problem is generated in the function getInput. Compared to Listing 1, the following changes need to be made:

• Remove Model.nu and Model.Condition, as Poisson ratio and plane-stress/strain condition are not required for diffusion problem.

• Replace Model.E with Model.d (i.e., change the definition of Young’s modulus $E$ to be the diffusion coefficient $d$).

• Replace Model.ElasticTensor with Model.DiffusiveTensor (i.e., change the definition of elastic tensor $\varvec{C}$ to be the diffusive tensor $\varvec{D}$).

• Set Model.DiffusiveTensor=diag([Model.d,Model.d])to define the diffusive tensor $\varvec{D} = \left[ {\begin{array}{*{20}l} d & 0 \\ 0 & d \\ \end{array} } \right]$.

• Set Model.DOFu=1 to change the nodal degrees of freedom DOFu from 2 to 1.

• Set u_exact=0.1+0.1*x1+0.2*x2 to define the exact solution $u^{\text{exact}}$.

In addition, we also need to modify the subroutine getBoundaryConditions to generate the exact boundary flux $t =\varvec{\eta}^{\text{T}} \varvec{D}{\varvec{\nabla}} u^{\text{exact}}$, source term $b = {\varvec{\nabla}} \cdot \left( {\varvec{D}{\varvec{\nabla}} u^{\text{exact}} } \right)$, essential boundary conditions $g = u^{\text{exact}}$, and switch matrix $S = 1$ based on a given expression of the exact solution $u^{\text{exact}}$ in a symbolic form, as shown in Listing 13.

Listing 13 shows command lines of function to generate exact heat flux $t$, heat source $b$, imposed scalar field $g$, and switch matrix $S$ for diffusion problems.

Due to the change of dimensionality in the $\varvec{B}$ and $\varvec{\varPsi}$ matrices compared to the elasticity problem, modifications are made to MatrixAssmebly (Listing 10) as follows

• Set d=Model.d to define the diffusivity coefficient.

• Set D=Model.DiffusiveTensor to define the diffusivity from input files.

• Replace E with d (i.e., replace the Young’s modulus E with diffusion coefficient d).

• Replace C with D (i.e., replace elastic tensor C with diffusive tensor D).

• Set B=sparse(2,nP*DOFu).

• Set PSI=sparse(1,nP*DOFu).

• Modify the allocation of the B and PSI from shape function SHP and derivative SHPDX1, SHPDX2 as:

  • PSI=SHP(idx_nQuad,:);

  • B(1,:)=SHPDX1(idx_nQuad,:);

  • B(2,:)=SHPDX2(idx_nQuad,:);

• Set ETA=[n1; n2;]to define the surface normal $\varvec{\eta}$.

With the above-mentioned modifications, RKPM2D is converted to a program for solving diffusion problems. By comparing the original code for elasticity problems with the modified code for diffusion problems, one can see that only minimal code modifications are required. This capability of easy code extension is a unique feature of RKPM2D [37].