Data discovering of inverse Robin boundary conditions problem in arbitrary connected domain through meshless radial point Hermite interpolation

In this paper, a suitable method is presented to treat the partial derivative equations, especially the Laplace equation having the Robin boundary conditions. These equations come from classical physics, especially the branch of thermodynamics, and have an efficient role in the field of heat and temperature. Our motivation is to reset a harmonic data obtained from Robin’s conditions in the arbitrary plane domain particularly on its boundaries. The applied method is a nodal Hermite meshless collocation technique at which it is formed of radial basis functions to get out the shape functions which is the key to construct the local bases in the neighborhoods of the nodal points. Moreover, by taking into consideration the Hermite interpolation technique, we can impose the boundary conditions directly, the named technique is called “MRPHI,” meshless radial point Hermite interpolation, and it is done on some examples so that trustworthy results are obtained.

     

Turing models in the biological pattern formation through spectral meshless radial point interpolation approach

In the present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of pattern formation in nonlinear reaction diffusion systems. Firstly, we obtain a time discrete scheme by approximating the time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, to eliminate the nonlinearity, a simple predictor–corrector (P–C) scheme is performed. The effect of parameters and conditions are studied by considering the well-known Schnakenberg model.

     

Two-dimensional capillary formation model in tumor angiogenesis problem through spectral meshless radial point interpolation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to a mathematical model for two-dimensional capillary formation model in tumor angiogenesis problem. This is a natural continuation of capillary formation in tumor angiogenesis (Shivanian and Jafarabadi in Eng Comput 34:603–619, 2018), where the capillary (1D problem) has been considered. The mathematical model describes the progression of tumor angiogenic factor in a unit square space domain, namely the extracellular matrix. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, and then, we use the SMRPI approach to approximate the spatial derivatives. This approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The obtained numerical results show that the SMRPI provides high accuracy and efficiency with respect to the other classical methods in the literature.

     

基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化

针对用有限元法进行连续体结构拓扑优化时需不断重构网格来处理网格畸变和网格移动,且存在数值计算不稳定等问题,基于无网格径向点插值方法(Radial Point Interpolation Method,RPIM)对简谐激励下的连续体结构进行拓扑优化.选取节点的相对密度作为设计变量,以结构动柔度最小化为目标函数,基于带惩罚的各向同性固体微结构(Solid Isotropic Microstructure with Penalization,SIMP)模型建立简谐激励下的优化模型;采用伴随法求解得到目标函数的敏度分析公式;利用优化准则法求解优化模型.经典的二维连续体结构拓扑优化算例证明该方法的可行性和有效性.     

Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM)

In this paper the meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation. The meshless LRPIM is one of the “truly meshless” methods since it does not require any background integration cells. In this case, all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. A technique is proposed to construct shape functions using radial basis functions. These shape functions which are constructed by point interpolation method using the radial basis functions have delta function property. The time derivatives are approximated by the time-stepping method. In order to eliminate the nonlinearity, a simple predictor-corrector scheme is performed. Numerical results are obtained for various cases involving line and ring solitons. Also the conservation of energy in undamped sine-Gordon equation is investigated.     

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.     

Boundary value problems in stochastic optimal control of advertising

Temporal patterns for advertising include constant spending over time, decreasing spending over time and increasing spending over time. This research shows that all these spending patterns emerge at optimality for the same response function dynamics, due to differences in salvage value assumptions. I use these results to develop a methodology for determining the optimal planning horizon length for each pattern of spending.     

Computer approach to the R-functions method of solution of boundary value problems in arbitrary domains

An analytically numerical method of solution of boundary-value problems is considered in arbitrary domains that may be concave and/or multiconnected. An essential feature of this so-called R-functions method (RFM) is a conversion of logical operations performed on sets (relevant to subdomains of which the considered domain is composed) into algebraic operations performed on elementary functions. The solution by the RFM is realized in two phases. In the first phase, an analytical formula for the so-called “general structure of solution” (GSS) is derived. GSS is a mapping that still contains undetermined function(s) but exactly satisfies all the prescribed boundary conditions. In the second phase, which is usually of numerical character, such function(s) is approximately evaluated by means of any suitable discrete method in order to satisfy the governing differential equation, which we consider or to minimize a relevant functional.Numerous tedious analytical operations, especially differentiations of complicated elementary functions, are necessary to derive GSS. In the original version of the method these had to be manually performed. This prevented many potential users from applying the RFM. Thus the main object of this work is to use the symbolic programming in order to obtain a fully computerized approach to the R-functions method. Both GSS itself and the results of all required operations performed on it are automatically obtained by the computer in an analytical form and written as FORTRAN subroutines ready for use in calculations. The Tschebychev approximation of undetermined function(s) and the least squares procedure complete this approach. As implemented only a few simple dates are required from the user. Some numerical examples are presented and discussed here. Suggestions are made as to areas of further research.The RFM may be applied to a wide class of linear and nonlinear boundary value problems in mechanics with the linear boundary conditions.     

Numerical solution of singularly perturbed two point boundary value problems by spline in compression

Some difference schemes for singularly perturbed two point boundary value problems are derived using spline in compression. These schemes are second order accurate. Numerical examples are given in support of the theoretical results.     

On the use of reduced models obtained through identification for feedback optimal control problems in a heat convection–diffusion problem

This paper deals with the use of reduced models for solving some optimal control problems. More precisely, the reduced model is obtained through the modal identification method. The test case which the algorithms is tested on is based on the flow over a backward-facing step. Though the reduction for the velocity fields for different Reynolds numbers is treated elsewhere [1], only the convection–diffusion equation for the energy problem is treated here. The model reduction is obtained through the solution of a gradient-type optimization problem where the objective function gradient is computed through the adjoint-state method. The obtained reduced models are validated before being coupled to optimal control algorithms. In this paper the feedback optimal control problem is considered. A Riccati equation is solved along with the Kalman gain equation. Additionally, a Kalman filter is performed to reconstruct the reduced state through previous and actual measurements. The numerical test case shows the ability of the proposed approach to control systems through the use of reduced models obtained by the modal identification method.