Purpose

Two-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.

Design/methodology/approach

The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.

Findings

The main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.

Originality/value

The originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

中文翻译：

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各向异性介质时空变系数非定常修正亥姆霍兹方程的边界元法和拉普拉斯变换组合

目的

二维 (2D) 问题受时空相关系数的非稳态各向异性修正亥姆霍兹方程控制。这些问题被转化为仅边界积分方程，可以使用标准边界元法 (BEM) 进行数值求解。通过算例验证了分析的有效性并检验了数值方法的准确性。

设计/方法论/途径

由时空相关系数的非稳态各向异性修正亥姆霍兹方程控制的二维问题可以使用边界元法和拉普拉斯变换的组合来求解。使用解析变换将时间-空间相关系数方程简化为时间相关系数方程。然后，对时变系数方程进行拉普拉斯变换，得到常系数方程，该方程可以写成仅边界积分方程。通过利用边界元法，求解该积分方程以找到拉普拉斯变换框架中问题的数值解。然后对这些解进行数值逆变换，以获得原始时空框架中的解。

发现

这项研究的主要发现是推导了仅边界积分方程，用于解决具有时间相关特性的各向异性功能梯度材料的时空相关系数的修正亥姆霍兹方程控制的初始边界值问题。

原创性/价值

该研究的原创性在于所考虑的功能梯度材料的性能的时间依赖性。