A meshless method using local exponential basis functions with weak continuity up to a desired order

Abstract

This paper introduces a novel meshless method based on the local use of exponential basis functions (EBFs). The EBFs are found so that they satisfy the governing equations within a series of subdomains. The compatibility between the subdomains is weakly satisfied through the minimization of a suitable norm written for the residuals of the continuity conditions. The residual norm may contain any desirable order of continuity. This allows increasing the continuity of the solution without increasing the type of point-wise variables at each node. The solution procedure begins with the discretization of the solution domain into a set of nodal points and cloud construction on each nodal point. The approximate solution in the local coordinates of each cloud is constructed by a series of EBFs. A set of intermediate points are distributed throughout the domain and its boundary to apply the continuity between the local solutions of the adjacent clouds up to a desired order, and also to impose the boundary conditions. The main nodes may play the role of the intermediate points as well. The validity of the results is shown through some patch tests. Also some numerical examples are solved to illustrate the capabilities of the method. High convergence rate of the numerical results is one of the salient features of the proposed meshless method.

Appendices

Appendix 1: Sampling theorem

Homogeneous basis functions have considerable effects on the accuracy of the results and therefore they should be selected properly. In order to investigate these basis functions more precisely, here we consider an EBF with a complex exponent in 1D as below

$$\begin{aligned} f\left( x \right) =e^{\alpha x}=e^{\left( {a+\hbox {i} b} \right) x}. \end{aligned}$$

(75)

Using Euler’s formula, it can be represented as

$$\begin{aligned} f\left( x \right) =e^{a x}\left( {\cos \left( {bx} \right) +\hbox {i} \sin \left( {bx} \right) } \right) . \end{aligned}$$

(76)

Thus the imaginary part of the exponent $\alpha$ displays the fluctuation frequency of the basis function. On the other hand the fluctuation frequencies of the homogeneous basis functions should be limited to an appropriate range in order to produce accurate results. We have employed the sampling theorem as a basis to determine this range. According to the sampling theorem, a band-limited signal can be reconstructed exactly if it is sampled at a rate at least twice the maximum frequency in it. It means that if the maximum frequency of a signal is $\omega _m$, then the sampling frequency should be more than $2\omega _m$, that is

$$\begin{aligned} \omega _s \ge 2\omega _m . \end{aligned}$$

(77)

Consequently by considering the nodal grid as the sampling points, the maximum frequency of the homogeneous basis functions is determined as below

$$\begin{aligned} \omega _m \le \frac{\pi }{L}, \end{aligned}$$

(78)

where $L$ is the distance between nodes in $x$ or $y$ directions.

Appendix 2: Weight factors

The residual formation in this method involves selection of the weight factors $w_u^i$ and $\mathbf{w}_\theta ^i$ for internal points and $w_N^i$ and $w_D^i$ for Neumann and Dirichlet boundary points. These weight factors may preferably be selected so that the terms in the total residual are of similar dimensions. Also it should be noted that the effect of the weight factors on the results comes from their relative values. Based on this, we consider $w_u^i =1$ and the other weight factors are determined accordingly.

The components of $\mathbf{w}_\theta ^i$ in (33) are multiplied by derivatives of the function $u$. On the other hand $u$ is approximated using EBFs and therefore its derivatives are dependent on the exponents. Being too small or too large, these exponents might have undesirable effects on the residual. These effects should be balanced by $\mathbf{w}_\theta ^i$. For this purpose, the components of $\mathbf{w}_\theta ^i$ are considered as below

$$\begin{aligned}&\mathbf{w}_\theta ^i =\left[ {{\begin{array}{ll} {w_{\theta x}^i }&{} 0 \\ 0&{} {w_{\theta y}^i } \\ \end{array} }} \right] , \quad w_{\theta x}^i =\frac{\sum _{l=1}^{m^{h}} {\left| {f_l^h \left( \mathbf{0} \right) } \right| } }{\sum _{l=1}^{m^{h}} {\left| {\partial _\mathrm{x} f_l^h \left( \mathbf{0} \right) } \right| } },\nonumber \\&\quad w_{\theta y}^i =\frac{\sum _{l=1}^{m^{h}} {\left| {f_l^h \left( \mathbf{0} \right) } \right| } }{\sum _{l=1}^{m^{h}} {\left| {\partial _\mathrm{y} f_l^h \left( \mathbf{0} \right) } \right| } }, \end{aligned}$$

(79)

where $\left| { \cdot } \right|$ is the absolute value function. In the above relation $f_l^h \left( \mathbf{0} \right) \equiv f_l^h |_{x=0, y=0}$ and $\partial _\varsigma f_l^h \left( \mathbf{0} \right) \equiv \partial _\varsigma f_l^h |_{x=0, y=0}$, ($\varsigma =x , y$).

The weight factor of the Neumann boundary condition $w_N^i$ can also be determined similar to $\mathbf{w}_\theta ^i$ with two differences. First, the derivatives in the denominators of the above fractions are replaced by their counterpart $\partial _\mathrm{n} f_l^h \left( \mathbf{0} \right)$ and second, because of the importance of the Neumann boundary conditions these fractions should be multiplied by a magnifying coefficient. Therefore this weight factor is suggested as below

$$\begin{aligned} w_N^i =S\times \frac{\sum _{l=1}^{m^{h}} {\left| {f_l^h \left( \mathbf{0} \right) } \right| } }{\sum _{l=1}^{m^{h}} {\left| {\partial _\mathrm{n} f_l^h \left( \mathbf{0} \right) } \right| } },\quad S=\left( {10^{0}\sim 10^{2}} \right) . \end{aligned}$$

(80)

Satisfaction of the Dirichlet boundary conditions is of great importance in the present method. Therefore the related terms in the residual should have greater weight factors. The author’s experiences show that choosing $w_D^i$ in the following range usually leads to accurate results

$$\begin{aligned} w_D^i =\left( {10^{2}\sim 10^{5}} \right) \times w_u^i. \end{aligned}$$

(81)

Appendix 3

To find polynomials satisfying the Laplace equation, $\nabla ^{2}u=0$, a polynomial of maximum order $m$ is considered as (56) which is repeated here as below

$$\begin{aligned} u_{ex} =\sum _{l=1}^M {a_l p_l } =\mathbf{P}\left( \mathbf{x} \right) \mathbf{a},\quad M=\left( {m+1} \right) \left( {m+2} \right) /2. \end{aligned}$$

(82)

Substituting $u_{ex}$ in the Laplace equation leads to

$$\begin{aligned} \nabla ^{2}u_{ex} =\sum _{l=1}^M {a_l \nabla ^{2}\left( {p_l } \right) } \equiv 0. \end{aligned}$$

(83)

Satisfaction of the above equation requires that a series of combinations of the monomials with similar orders vanish. This results in a set of $\bar{{N}}$ equations as below

$$\begin{aligned}&\left[ {{\begin{array}{cccc} {A_{11} }&{} \quad {A_{12} }&{}\quad \cdots &{}\quad {A_{1M} } \\ {A_{21} }&{}\quad {A_{21} }&{}\quad \cdots &{}\quad {A_{2M} } \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ {A_{\bar{{N}}1} }&{}\quad {A_{\bar{{N}}2} }&{} \quad \cdots &{} \quad {A_{\bar{{N}}M} } \\ \end{array} }} \right] \left\{ {{\begin{array}{c} {a_1 } \\ {a_2 } \\ \vdots \\ {a_M } \\ \end{array} }} \right\} =\left\{ {{\begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ \end{array} }} \right\} ,\nonumber \\&\quad \bar{{N}}=\frac{m\left( {m-1} \right) }{2}<M=\frac{\left( {m+1} \right) \left( {m+2} \right) }{2}. \end{aligned}$$

(84)

Therefore we have $\bar{{N}}$ equations and $M$ coefficients which means that the number of independent coefficients is $M-\bar{{N}}$. Now by considering a set of independent coefficients as below

$$\begin{aligned} {\bar{\mathbf{a}}}=\left\{ {{\begin{array}{llll} {\bar{{a}}_1 }&{} {\bar{{a}}_2 }&{} \ldots &{} {\bar{{a}}_{M-\bar{{N}}} } \\ \end{array} }} \right\} ^{T}, \end{aligned}$$

(85)

all the coefficients can be expressed as

$$\begin{aligned} \begin{aligned}&\left( \mathbf{a} \right) _{M\times 1} =\left( {\bar{\mathbf{A}}} \right) _{M\times \left( {M-\bar{{N}}} \right) } \left( {\bar{\mathbf{a}}} \right) _{\left( {M-\bar{{N}}} \right) \times 1},\\&\quad a_l =\sum _{j=1}^{M-\bar{{N}}} {\bar{{A}}_{lj} \bar{{a}}_j } ,\quad l=1,\ldots ,M. \end{aligned} \end{aligned}$$

(86)

Finally by inserting (86) into (82), we have

$$\begin{aligned} u_{ex}&= \sum _{l=1}^M {a_l p_l } =\sum _{l=1}^M {\left( {\sum _{j=1}^{M-\bar{{N}}} {\bar{{A}}_{lj} \bar{{a}}_j } } \right) p_l }\nonumber \\&= \sum _{j=1}^{M-\bar{{N}}} {\left( {\sum _{l=1}^M {\bar{{A}}_{lj} p_l } } \right) \bar{{a}}_j } =\sum _{j=1}^{M-\bar{{N}}} {\bar{{p}}_j \bar{{a}}_j } , \end{aligned}$$

(87)

where $M-\bar{{N}}=2m+1$. For instance, if we consider a polynomial of the order $m=6$, the homogeneous counterpart is obtained as below

$$\begin{aligned} m=6\rightarrow M=28,\quad \bar{{N}}=15,\quad M-\bar{{N}}=13. \end{aligned}$$

(88)

This means that the original exact solution is considered as

$$\begin{aligned} u_{ex} =\sum _{l=1}^{28} {a_l p_l } =a_1 +a_2 x+a_3 y+a_4 x^{2}+\cdots +a_{28} y^{6}, \end{aligned}$$

(89)

which converts to

$$\begin{aligned} u_{ex}&= \sum _{j=1}^{13} {\bar{{a}}_j \bar{{p}}_j } =\bar{{a}}_1 +\bar{{a}}_2 x+\bar{{a}}_3 y+\bar{{a}}_4 xy+\bar{{a}}_5 \left( {x^{2}-y^{2}} \right) \nonumber \\&\quad +\,\bar{{a}}_6 \left( {x^{3}-3xy^{2}} \right) +\bar{{a}}_7 \left( {y^{3}-3x^{2}y} \right) \nonumber \\&\quad +\,\bar{{a}}_8 \left( {x^{4}-6x^{2}y^{2}+y^{4}} \right) +\bar{{a}}_9 \left( {x^{3}y-xy^{3}} \right) \nonumber \\&\quad +\, \bar{{a}}_{10} \left( {x^{5}-10x^{3}y^{2}+5xy^{4}} \right) \nonumber \\&\quad +\,\bar{{a}}_{11} \left( {y^{5}-10x^{2}y^{3}+5x^{4}y} \right) \nonumber \\&\quad +\, \bar{{a}}_{12} \left( {x^{6}-15x^{4}y^{2}+15x^{2}y^{4}-y^{6}} \right) \nonumber \\&\quad +\,\bar{{a}}_{13} \left( {3x^{5}y+3xy^{5}-10x^{3}y^{3}} \right) . \end{aligned}$$

(90)

Now the polynomials $\bar{{p}}_j$ are considered in place of $p_l$ in (57) to perform the patch test.