Timestep-dependent element interpolation functions in the method of matched sections on the example of heat conduction problem

Abstract

The paper is devoted to further elaboration of the method of matched sections as a new technique within the finite element method. Like FEM it supposes that: a) the complex domain is represented as a mesh of nonintersecting simple elements; b) algebraic relations between the main parameters of an element are established from the governing differential equations; c) all relationships from all elements are assembled into one global matrix. On the other hand, it has two distinct features. The first one is that relations between kinematic and inner force parameters (called Connection equations) are derived from the approximate analytical solution of the governing equations rather than by the application of minimization techniques. The second one consists in that the conjugation between elements is provided between the adjacent sides (sections) rather than in the nodes of the elements. In application to the transient 2D heat conduction, it is assumed that for each small rectangular element, the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, $T$, and heat flux $Q$. In practical realization for rectangular finite elements, the method is reduced to the determination of eight unknowns for each element – two unknowns on each side, which are related by the connection equations, and the requirement of the temperature continuity at the center of the element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step becomes the parameter of the element interpolation function within the element, i.e. it determines the behavior of the connection equations. This method was initially proposed by the first author for several 1D problems, and here for the first time, it is applied to 2D problems. The number of tests for rectangular plates exhibits the remarkable properties of the proposed time integration scheme concerning stability, accuracy, and absence of any restrictions as to increasing the time step.