# Thermal Accelerometer Simulation by the R-Functions Method
## 1 Introduction  
Many modern devices, in particular, sensors based on microelectromechanical systems (MEMS), need a sophisticated mathematical simulation to find the ways of their optimization. Among them are thermal accelerometers (TAs) [1] which principle of operation is based on the effect of fluid or gas convection in closed cavities under the influence of external forces of inertia. It is very important to estimate the scale factor and bias stabilities of TAs under an external thermal slope, and to test different types of cavity geometry (cylindrical, rectangular, hexagonal, etc.) to achieve the best performance of the device. To solve the aforementioned problems, it is necessary to simulate diffusive and convective heat and mass transfer in arbitrarily shaped enclosures, which in turn is a problem of great importance due to a large number of its practical applications. Since analytical solutions exist for a very narrow class of simple-shaped domains only, a majority of modern computational fluid dynamics (CFD) techniqueswere developed which are based mainly on the use of the finite-difference and finite- element methods (FDM and FEM) [2; 3; 4], boundary-element methods [5], and spectral element methods [6]. Their main drawback is the cumbersome representation of grid solutions in complex- shaped domains, as well as the difficulties connected with the approximation of boundary conditions. For the function of pressure or for the vorticity function in dimensionless form, boundary conditions are not specified at all and require special approximations. In addition, sometimes it is difficult to interpret analytically the solutions obtained at a number of mesh knots and to evaluate errors. Recently, a number of meshless (or mesh-free) techniques were proposed for solving convection-diffusion problems with arbitrarily shaped cavities [7; 8; 9]. They became popular due to their simplicity, flexibility, and independence from a complicated domain geometry. One of the most effective meshless techniques is the [PERSON]'s R-function method (RFM) [10]. It allows the given boundary conditions to be satisfied exactly at all boundary points by means of the appropriate transformation of the basic functions. With R-functions, it became possible to construct the functions with prescribed values and derivatives at specified locations. Furthermore, the constructed functions possess desired differential properties and may be assembled into a solution structure for the posed boundary value problems. The semi-analytical method of R-functions takes an intermediate position between traditional analytical approaches and numerical tools. Many practical problems in different areas of mathematical physics, in particular, heat conductivity [11], are being solved effectively by using RFM. Application of the RFM in combination with the Galerkin technique for solving CFD problems in arbitrarily shaped domains was investigated earlier [12; 13; 14]. Different bases were applied in these works, both spectral (polynomial) and local (B-splines), and good results were achieved for fields evaluated in domains of simple geometry without localized inhomogeneities. In [15], an approach combining the RFM and the Petrov-Galerkin method with bases of algebraic polynomials was presented and applied to the simulation of thermal convection fields inside a closed rectangular cavity. However, when applying the RFM, if a domain consists of two parts, for example, \"inner\" and \"outer\", which characteristic dimensions are not comparable, one needs to take either a large higher power of approximating polynomials (in spectral methods) that worsens computational stability, or a very small regular or irregular mesh width for bases of compactly supported functions. Therefore, in the latter case, the RFM loses its main benefits and can be practically considered as a conventional grid method such as FDM or FEM. Here, a novel technique is proposed (MQ-RFM), based on the RFM with multiquadric (MQ) radial basis functions (RBFs) [16; 17].