Meshless Local Strong Form Method (MLSM)
The Meshless Local Strong Form Method (MLSM) is a generalization of methods which are in literature also known as Diffuse Approximate Method (DAM), Local Radial Basis Function Collocation Methods (LRBFCM), Generalized FDM (GFDM), Collocated discrete least squares (CDLS) meshless, etc. Although each of the named methods has some unique properties, the basic concept of all local strong form methods is similar, namely to approximate differential operators as
weighted sums of function values over the local support domain. MLSM can be understood as a generalization of FDM to scattered nodes, however much more powerful.
The elegance of MLSM is its simplicity and generality. The presented methodology can be also easily upgraded or altered, e.g. with nodal adaptation, basis augmentation, conditioning of the approximation, etc., to treat anomalies such as sharp discontinues or other obscure situations, which might occur in complex simulations. In MLSM, the type of approximation, the size of support domain, and the type and number of basis function is general. For example, minimal support size for 2D transport problem is five, however higher support domains can be used to stabilize computations on scattered nodes at the cost of computational complexity. Various types of basis functions might appear in the approximation, however, the most commonly used are multiquadrics, Gaussians and monomials. Some authors also enrich the radial basis with monomials to improve performance of the method (see [[]]). All these features can be controlled on the fly during the simulation. From the computation point of view, the localization of the method reduces inter-processor communication, which is often a bottleneck of parallel algorithms.
The core of the spatial discretization is a local Moving Least Squares (MLS) approximation of a considered field over the overlapping local support domains, i.e. in each node we use approximation over a small local sub-set of neighbouring $n$ nodes. The trial function is thus introduced as \[\hat{u}(\mathbf{p})=\sum\limits_{i}^{m}{{{\alpha }_{i}}{{b}_{i}}(\mathbf{p})}=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}\mathbf{\alpha }\] with $m,\,\,\mathbf{\alpha }\text{,}\,\,\mathbf{b},\,\,\mathbf{p}\left( {{p}_{x}},{{p}_{y}} \right)$ standing for the number of basis functions, approximation coefficients, basis functions and the position vector, respectively.
The problem can be written in matrix form (refer to Moving Least Squares (MLS) for more details) as \[~\mathbf{\alpha }={{\left( {{\mathbf{W}}^{0.5}}\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\mathbf{u}\] where $(\mathbf{W}^{0.5}\mathbf{B})^{+}$ stand for a Moore–Penrose pseudo inverse.
Shape functions
Refer to Moving Least Squares (MLS) and Computation of shape functions for details about numerical calculation of shape functions.
By explicit expressing the coefficients $\alpha$ into the trial function \[~\hat{u}\left( \mathbf{p} \right)=\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{u}=\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\] is obtained, where $\mathbf{\chi}$ stand for the shape functions. Now, we can apply partial differential operator, which is our goal, on the trial function \[L~\hat{u}\left( \mathbf{p} \right)=L\mathbf{\chi }\left( \mathbf{p} \right)\mathbf{u}\] where $L$ stands for general differential operator. For example: \[{{\mathbf{\chi }}^{\partial x}}\left( \mathbf{p} \right)=\frac{\partial }{\partial x}~\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] \[{{\mathbf{\chi }}^{\partial y}}\left( \mathbf{p} \right)=\frac{\partial }{\partial y}~\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] \[{{\mathbf{\chi }}^{{{\nabla }^{2}}}}\left( \mathbf{p} \right)={{\nabla }^{2}}\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\]
The presented formulation is convenient for implementation since most of the complex operations, i.e. finding support nodes and building shape functions, are performed only when nodal topology changes. In the main simulation, the pre-computed shape functions are then convoluted with the vector of field values in the support to evaluate the desired operator. The presented approach is even easier to handle than the FDM, however, despite its simplicity it offers many possibilities for treating challenging cases, e.g. nodal adaptivity to address regions with sharp discontinuities or $p$-adaptivity to treat obscure anomalies in physical field, furthermore, the stability versus
computation complexity and accuracy can be regulated simply by changing number of support nodes, etc. All these features can be controlled on the fly during the simulation by re computing the shape functions with different setups. However, such re-setup is expensive, since the \[\mathbf{b}{{\left( \mathbf{p} \right)}^{\text{T}}}{{\left( {{\mathbf{W}}^{0.5}}\left( \mathbf{p} \right)\mathbf{B} \right)}^{+}}{{\mathbf{W}}^{0.5}}\] has to be re-evaluated, with asymptotical complexity of $O\left( {{N}_{D}}n{{m}^{2}} \right)$, where ${{N}_{D}}$ stands for total number of discretization nodes. In addition, the determination of support domain nodes also consumes some time, for example, if a kD-tree data structure is used, first the tree is built with $O\left( {{N}_{D}}\log {{N}_{D}} \right)$ and then additional $O\left( {{N}_{D}}\left( log{{N}_{D}}+n \right) \right)$ for collecting $n$ supporting nodes.
References
- Kosec G., A local numerical solution of a fluid-flow problem on an irregular domain. Advances in engineering software. 2016;7 ; [29512743] :: manuscript
- Trobec R., Kosec G., Šterk M., Šarler B., Comparison of local weak and strong form meshless methods for 2-D diffusion equation. Engineering analysis with boundary elements. 2012;36:310-321; manuscript
- Kosec G, Sarler B. Solution of thermo-fluid problems by collocation with local pressure correction. International Journal of Numerical Methods for Heat & Fluid Flow. 2008;18:868-82 manuscript
- Trobec R., Kosec G., Parallel Scientific Computing, ISBN: 978-3-319-17072-5 (Print) 978-3-319-17073-2.
