Meshless numerical analysis of partial differential equations with nonlinear inequality constraints
- Mei Chen^{1} and
- Xiaolin Li^{1}Email author
https://doi.org/10.1186/s13662-015-0514-7
© Chen and Li 2015
Received: 28 January 2015
Accepted: 20 May 2015
Published: 10 June 2015
Abstract
A meshless method for the numerical solution of partial differential equations with nonlinear inequality constraints is discussed in this paper. The original nonlinear inequality problem is linearized as a sequence of linear equality problems, and then discrete linear system of algebraic equations is formed. This meshless method only requires nodes on the boundary of the domain, and it does not require any numerical integrations. Numerical experiments indicate that this method is very effective for nonlinear inequality problems and has good convergence rate and high computational efficiency.
Keywords
1 Introduction
The inequality constraint (4) is nonlinear and complementary. The nonlinear inequality problem (1)-(4) arises from many kinds of physical and industrial applications such as the groundwater flow problem [1, 2], the electropaint process [3, 4], the contact problem [5], the free surface problem [6] and the variational inequalities theory [7, 8]. The numerical solution of this kind of problems can be obtained using the finite difference method and the finite element method (FEM) [1, 3, 4]. However, these domain-type numerical techniques require domain meshing which is usually arduous and computationally expensive.
In problem (1)-(4), u and q alternate on \(\Gamma_{R} \) in conjunction with nonlinear inequality constraint. To obtain the solution u in Ω, we first need to determine on which parts of \(\Gamma_{R} \) the boundary conditions \(u=f\) and \(q< g\) apply, and thus on the remaining parts the boundary conditions \(u< f\) and \(q=g\) apply. Therefore, the primary focus in solving this problem is on \(\Gamma_{R} \) and thus, boundary-type numerical methods such as the boundary element method (BEM) [5, 6, 9–12] are particularly suitable for the solution of such problems. The BEM involves the generation of elements on the boundary surface and the computation of some complex singular integrals on boundary elements. In some cases, these processes can also be very difficult and computationally expensive, especially for free boundary and nonlinear problems.
Boundary-type numerical methods reduce the computational dimensions of the original problem by one and thus simplify the efforts involved in data preparation and CPU time. In the past two decades, meshless (or meshfree) methods for numerical solutions of partial differential equations have been developed for overcoming the meshing-related drawbacks involved in the FEM and the BEM. Some boundary-type meshless methods using meshless shape functions and boundary integral equations have been developed. Among them are the boundary node method (BNM) [12, 13], the boundary point interpolation method (BPIM) [14, 15], the hybrid BNM [16, 17], the Galerkin BNM [18, 19], the dual BNM [20, 21] and the boundary element-free method [22–24]. These boundary-type meshless methods perform very well for the numerical solution of lots of linear problems. However, maybe due to the issues associated with the handling of the nonlinear inequality constraints, not many boundary-type meshless methods have been used to the nonlinear inequality problems. Besides, these boundary-type meshless methods still involve the computation of complex singular boundary integrals.
The aim of this paper is to develop a boundary-type meshless method for the numerical solution of the nonlinear inequality problem (1)-(4). This method only requires boundary nodes, and does not need any mesh for either interpolation or integration. Besides, no integrations are involved in the whole solution process. Namely, the present method is truly meshless, integration-free and boundary-only. As a result, this method is expected to have higher computational speed and efficiency. Numerical experiments indicate that this method is very effective for problems with nonlinear inequality constraints and has good convergence rate and high computational efficiency. The following discussions begin with a linearization of the nonlinear inequality problem in Section 2. Then, a detailed meshless numerical implementation is presented in Section 3. Numerical experiments are provided in Section 4. Finally, Section 5 contains some conclusions.
2 Linearization
Eq. (4) indicates that \(q\le g\). If \(q=g\), then Eq. (5) indicates that \(( q-g )-c ( u-f )\ge0\), and thus \(u\le f\). Otherwise, if \(q< g\), then \(( q-g )-c (u-f)<0\), and thus recalling again Eq. (5) leads to \(u=f\). As a result, from Eq. (5) we can deduce the nonlinear inequality constraint (4). On the other hand, from constraint (4) we can deduce Eq. (5) immediately. Summarizing, we have shown that the nonlinear inequality constraint (4) is equivalent to Eq. (5).
3 Meshless numerical implementation
An integration-free and truly meshless method is developed in this section for the nonlinear inequality problem (1)-(4).
To obtain the unknown coefficient \(\eta_{j}^{ ( k )} \) and to simplify the representation, we assume that the first \(N_{d} \) boundary nodes \(\{ {\mathbf{x}_{i} } \}_{i=1}^{N_{d} } \) belong to \(\Gamma_{D} \), the next \(N_{n} \) boundary nodes \(\{ {\mathbf{x}_{i} } \}_{i=N_{d} +1}^{N_{d} +N_{n} } \) belong to \(\Gamma_{N} \) and the remaining \(N_{r} =N-N_{d} -N_{n} \) boundary nodes \(\{ {\mathbf{x}_{i} } \}_{i=N_{d} +N_{n} +1}^{N} \) belong to \(\Gamma_{R} \).
Once \(u^{ ( 0 )} (\mathbf{x}_{i} )\) and \(q^{ ( 0 )} (\mathbf{x}_{i} )\) are found, the linear problem (10)-(13) can be solved iteratively. The details follow.
- 1.
Choose a tolerance \(\varepsilon>0\) and N boundary nodes \(\{ {\mathbf{x}_{i} } \}_{i=1}^{N} \subset\Gamma\), and obtain \(\{ {\mathbf{y}_{j} } \}_{j=1}^{N} \) using Eq. (22).
- 2.
Form the linear algebra system given by Eqs. (25)-(27), solve the system to obtain \(\eta_{j}^{ ( 0 )} \), and compute \(u^{ ( 0 )} (\mathbf{x}_{i} )\) and \(q^{ ( 0 )} (\mathbf{x}_{i} )\) using Eq. (28) for all nodes \(\mathbf{x}_{i} \in\Gamma_{R} \).
- 3.
Compute A, \(\bar{\mathbf{u}}\), \(\bar{\mathbf{q}}\) and K, and set \(k=1\).
- 4.
Compute \(\mathbf{r}^{ (k-1)}\) and form \(\mathbf{b}^{ (k-1)}\), obtain \(\boldsymbol{\eta}^{ ( k )}\) by computing \(\mathbf{Kb}^{ (k-1)}\), and then obtain \(u^{ ( k )} (\mathbf{x}_{i} )\) and \(q^{ ( k )} (\mathbf{x}_{i} )\) using Eq. (37) for all nodes \(\mathbf{x}_{i} \in\Gamma_{R} \).
- 5.
If \(e ( u ):=\sqrt{{\sum_{\mathbf{x}_{i} \in\Gamma_{R} } { ( u^{ ( k )} (\mathbf{x}_{i} )-u^{ (k-1)} (\mathbf{x}_{i} ) )^{2}} } / {\sum_{\mathbf{x}_{i} \in\Gamma_{R} } ( u^{ ( k )} (\mathbf{x}_{i} ) )^{2} }} \le\varepsilon\), stop the iterative process and return \(\boldsymbol{\eta}^{ ( k )}\) for computing \(u^{ ( k )} ( \mathbf{x} )\).
- 6.
Otherwise, update k to \(k+1\), and go to step 4.
The iterative process continues until the stopping criterion \(e ( u )\le\varepsilon\) is satisfied. In the subsequent numerical examples, the tolerance is taken as \(\varepsilon=10^{-6}\). Obviously, the number of iteration is determined uniquely by the stopping criterion.
From the above discussion, we can conclude that no elements are used and no integrations are computed in the whole iterative process. Thus, the present numerical method is integration-free and truly meshless.
4 Numerical experiments
4.1 Analysis of accuracy and convergence of our method
Number of iteration
N | 32 | 64 | 128 | 256 | 512 | 1,024 |
---|---|---|---|---|---|---|
BEM | 56 | 91 | 141 | 203 | 263 | 319 |
Our method | 20 | 28 | 37 | 48 | 54 | 47 |
CPU time (in seconds)
N | 32 | 64 | 128 | 256 | 512 | 1,024 |
---|---|---|---|---|---|---|
BEM | 0.094 | 0.359 | 0.796 | 4.274 | 31.839 | 182.426 |
Our method | 0.031 | 0.063 | 0.141 | 0.624 | 3.885 | 6.958 |
4.2 Groundwater flow problem
CPU time (in seconds) for the groundwater flow problem
N | 32 | 64 | 128 | 256 | 512 | 1,024 |
---|---|---|---|---|---|---|
BEM | 0.188 | 0.328 | 2.012 | 14.274 | 93.069 | 673.173 |
Our method | 0.047 | 0.078 | 0.156 | 0.499 | 2.043 | 8.814 |
4.3 An electropaint process
CPU times (in seconds) for the electropainting problem
N | 120 | 240 | 480 | 960 | 1,920 |
---|---|---|---|---|---|
BEM | 1.404 | 4.867 | 20.499 | 91.403 | 462.795 |
Our method | 0.047 | 0.078 | 0.281 | 1.528 | 9.501 |
4.4 Problem in a half-oval torus domain
5 Conclusions
Meshless methods for partial differential equations with linear equality constraints are now well established. Unfortunately, presumably because nonlinearity and inequation simultaneously exist, the research on meshless methods for problems with nonlinear inequality constraints is scarce. In this paper, a meshless method is presented to deal with the numerical solution of partial differential equations with nonlinear inequality constraints. In this numerical method, the nonlinear inequality constraints are linearized naturally and efficiently, then the original nonlinear inequality problem is transformed into a sequence of linear equality problems. The meshless method in this paper inherently has some desirable numerical merits such as truly meshless, boundary-only and integration-free. Numerical examples are presented for some partial differential equations with nonlinear inequality constraints. The numerical results show that the present meshless method has the merits of good convergence rate, and higher computational accuracy and efficiency over the traditional mesh-based methods such as the BEM.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11471063), the Natural Science Foundation Project of CQ CSTC (No. cstc2014jcyjA00005) and the Talent Project of Chongqing Normal University (No. 14CSBJ04).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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