# Thermal Accelerometer Simulation by the R-Functions Method
## 6 Numerical Experiment  
Consider the model boundary value problem in circular annulus (Figure 3a) with boundary conditions:  
\[ \begin{array}{l}X^{2}+\ Y^{2}=R_{i}:\ \theta=1,\ \psi=\frac{\partial\psi}{ \partial\mathbf{n}}=U=V=0;\\ X^{2}+\ Y^{2}=R_{o}:\ \theta=\psi=\frac{\partial\psi}{\partial\mathbf{n}}=U=V=0. \end{array}  \tag{33}\]  
The common way to obtain a function describing the boundary of the domain shown in Figure 3b is the use of R-conjunction operation:  
\[\omega=\omega_{\mathrm{C}}\wedge\omega_{H}=\omega_{\mathrm{C}}+\omega_{H}- \sqrt{\omega_{\mathrm{C}}^{2}+\omega_{H}^{2}}. \tag{34}\]  
For the circular annulus, expressions for \"hot\" and \"cold\" parts of the boundary can be written as follows:  
\[\omega_{H}(X,Y)=-\frac{1}{2R_{\mathrm{i}}}\Big{(}R_{\mathrm{i}}^{2}-X^{2}-Y^ {2}\Big{)},\ \omega_{\mathrm{C}}(X,Y)=\frac{1}{2R_{\mathrm{o}}}\Big{(}R_{\mathrm{o}}^{2}- X^{2}-Y^{2}\Big{)}. \tag{35}\]  
Here, the normalizing terms \(1/(2R_{\mathrm{i}/o})\) are taken to provide the normalized functions \(\omega_{\mathrm{C}},\omega_{H}\) whose values, as well as values of \(\omega\), are close to the distances to corresponding boundaries at points located in their vicinity. At first glance, for such a simple domain as shown in Figure 3a, by analogy with the approach proposed in [38; 39], one would take instead of Equation (34) another expression:  
\[\omega=\omega_{\mathrm{C}}\omega_{H}=-\Big{(}R_{\mathrm{i}}^{2}-X^{2}-Y^{2} \Big{)}\Big{(}R_{\mathrm{o}}^{2}-X^{2}-Y^{2}\Big{)}. \tag{36}\]  
However, this way of constructing the function \(\omega\) is restricted by the narrow class of domains (eccentric circular annuli, elliptic annuli, etc.) and it is inappropriate, for example, for annular domains with more complicated inner boundaries (triangular, rectangular, et al. [44; 45]). Additionally, due to the fact that the function (36) is not normalized, namely,  
\[\frac{\partial\omega}{\partial\mathbf{n}}\Big{|}_{\partial\Omega_{\mathrm{C}} }=\left.\omega_{H}\right|_{\partial\Omega_{\mathrm{C}}},\ \frac{\partial\omega}{\partial\mathbf{n}}\Big{|}_{\partial\Omega_{H}}=\left. \omega_{\mathrm{C}}\right|_{\partial\Omega_{H}},\]  
we shall obtain more complicated formulae for the local Nusselt numbers than Equation (31) while the application of functions (35) yields the following expressions:  
\[ \begin{array}{l}\mathrm{Nul}|_{\partial\Omega_{\mathrm{C}}}=\sum\limits_{n= 0}^{N}c_{n}^{(k)}\chi_{n}|_{\partial\Omega_{\mathrm{C}}}+2R_{\mathrm{o}}\frac{ \theta_{H}-\partial_{\mathrm{C}}}{R_{\mathrm{o}}^{2}-R_{\mathrm{i}}^{2}},\ \mathrm{Nul}|_{\partial\Omega_{H}}=\sum\limits_{n=0}^{N}c_{n}^{(k)}\chi_{n}|_{ \partial\Omega_{H}}+2R_{\mathrm{i}}\frac{\partial_{\mathrm{C}}-\theta_{H}}{R_ {\mathrm{o}}^{2}-R_{\mathrm{i}}^{2}};\\ \overline{\mathrm{Nul}}|_{\partial\Omega_{\mathrm{C}}}=\sum\limits_{n=0}^{N}c _{n}^{(k)}\int\limits_{\partial\Omega_{\mathrm{C}}}\chi_{n}\mathrm{d}s+2R_{ \mathrm{o}}\frac{\theta_{H}-\partial_{\mathrm{C}}}{R_{\mathrm{o}}^{2}-R_{ \mathrm{i}}^{2}},\ \overline{\mathrm{Nul}}|_{\partial\Omega_{H}}=\sum\limits_{n=0}^{N}c _{n}^{(k)}\int\limits_{\partial\Omega_{H}}\chi_{n}\mathrm{d}s+2R_{\mathrm{i}} \frac{\partial_{\mathrm{C}}-\theta_{H}}{R_{\mathrm{o}}^{2}-R_{\mathrm{i}}^{2}}. \end{array} \]  
With the help of the transfinite interpolation (26), we pass to the problem with homogeneous boundary conditions:  
\[\Phi=\frac{\omega_{\mathrm{C}}}{\omega_{H}+\omega_{\mathrm{C}}}. \tag{37}\]  
In this work, we take the basis of multiquadric RBF (MQ-RBF) which is often used in various meshless techniques [18; 19]:  
\[\chi_{n}(X,Y)=\sqrt{\Big{(}X-X_{i(n)}\Big{)}^{2}+\Big{(}Y-Y_{j(n)}\Big{)}^{2}+ \alpha^{2}}, \tag{38}\]  
where \(\Big{(}X_{i(n)}Y_{j(n)}\Big{)}\) are the centers of functions \(\chi_{n}\) and \(\alpha\) is their shape parameter. For simplicity we take \(\Big{(}X_{i(n)}Y_{j(n)}\Big{)}\) as knots of a regular two-dimensional square mesh with widths \(h_{X}=h_{Y}=\alpha\).