Purpose

The study of heat transfer mechanisms is an area of great interest because of various applications that can be developed. Mathematically, these phenomena are usually represented by partial differential equations associated with initial and boundary conditions. In general, the resolution of these problems requires using numerical techniques through discretization of boundary and internal points of the domain considered, implying a high computational cost. As an alternative to reducing computational costs, various approaches based on meshless (or meshfree) methods have been evaluated in the literature. In this contribution, the purpose of this paper is to formulate and solve direct and inverse problems applied to Laplace’s equation (steady state and bi-dimensional) considering different geometries and regularization techniques. For this purpose, the method of fundamental solutions is associated to Tikhonov regularization or the singular value decomposition method for solving the direct problem and the differential Evolution algorithm is considered as an optimization tool for solving the inverse problem. From the obtained results, it was observed that using a regularization technique is very important for obtaining a reliable solution. Concerning the inverse problem, it was concluded that the results obtained by the proposed methodology were considered satisfactory, as even with different levels of noise, good estimates for design variables in proposed inverse problems were obtained.

Design/methodology/approach

In this contribution, the method of fundamental solution is used to solve inverse problems considering the Laplace equation.

Findings

In general, the proposed methodology was able to solve inverse problems considering different geometries.

Originality/value

The association between the differential evolution algorithm and the method of fundamental solutions is the major contribution.

中文翻译：

---

使用基本解和微分演化的方法解决正向和逆向传导传热问题

目的

由于可以开发各种应用，因此对传热机理的研究引起了极大的兴趣。在数学上，这些现象通常由与初始条件和边界条件相关的偏微分方程表示。通常，解决这些问题需要通过离散化所考虑域的边界和内部点来使用数值技术，这意味着较高的计算成本。作为减少计算成本的替代方法，在文献中已经评估了基于无网格（或无网格）方法的各种方法。在此贡献中，本文的目的是考虑到不同的几何形状和正则化技术，公式化并求解应用于拉普拉斯方程（稳态和二维）的正反问题。以此目的，基本解法与Tikhonov正则化或奇异值分解法有关，可以解决直接问题，而差分进化算法则被视为解决逆问题的优化工具。从获得的结果可以看出，使用正则化技术对于获得可靠的解决方案非常重要。关于反问题，得出的结论是，即使在噪声水平不同的情况下，通过所提出的方法所获得的结果也被认为是令人满意的，即使对于所提出的反问题也可以获得对设计变量的良好估计。

设计/方法/方法

在这一贡献中，基本解法被用于解决考虑拉普拉斯方程的反问题。

发现

通常，所提出的方法能够解决考虑不同几何形状的逆问题。

创意/价值

差分进化算法和基本解法之间的联系是主要的贡献。