# Generalization of a reduced Trefftz type approach

## TL;DR

This paper explores variational concepts of reduced Trefftz methods, analyzing their relationship with displacement and hybrid approaches, and proposes strategies for developing invariant finite elements with demonstrated convergence and performance benefits.

## Contribution

It introduces a unified variational framework for reduced Trefftz approaches and presents a new strategy for creating invariant finite elements with broad geometric applicability.

## Key findings

- Demonstrated convergence in linear static and kinetic problems
- Compared reduced Trefftz methods with other approaches for conformity
- Showed improved numerical performance of developed finite elements

## Abstract

Summary This work presents variational concepts associated with reduced Trefftz type approaches and discusses the interrelationship between various concepts of the displacement, hybrid and Trefftz methods. The basic concept of the displacement version of the reduced Trefftz method operates on the natural boundary conditions enforced in an integral form whereas the stress version of the reduced Trefftz type approach operates on the essential boundary conditions enforced in an integral sense. The application of the method proposed in the framework of the finite element method is briefly outlined. The methods used by the reduced Trefftz type approach for enforcing conformity and interelement continuity between neighboured elements are also discussed. Comparisons with other known methods for the same purpose are performed. General strategy for developing finite elements of general geometric form such as quadrilateral elements with invariance properties is presented. The basic idea of this strategy consists in using the natural coordinate system only for defining the element geometry and performing the element integration in the biunit interval. For defining the approximation functions a local coordinate system defined from the directions of the covariant base vectors and the perpendicular contravariant base vectors computed in the geometric centre of the element is used. This strategy can also be used to implement other versions of finite elements and other forms of finite elements. Different numerical calculations and comparisons in the linear statics and kinetics are performed in order to assess the convergence and the numerical performance of finite elements developed by applying the reduced Trefftz type approach.

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Source: https://tomesphere.com/paper/1706.07794