Trefftz method
Encyclopedia
In mathematics
, the Trefftz method is a method for the numerical solution
of partial differential equation
s named after the German
mathematician Erich Trefftz (1888-1937). It falls within the class of finite element method
s.
that governs the problem into a variational functional
from which element nodal properties – known as field variables – can be found. This can be solved by substituting in approximate solutions to the differential equation and generating the finite element stiffness matrix
which is combined with all the elements in the continuum to obtain the global stiffness matrix [2]. Application of the relevant boundary conditions to this global matrix, and the subsequent solution of the field variables rounds off the mathematical process, following which numerical computations can be used to solve real life engineering problems [1, 3].
An important aspect of solving the functional requires us to find solutions which satisfy the given boundary conditions and satisfy inter-element continuity
since we define independently the properties over each element domain
[1].
The hybrid Trefftz method differs from the conventional finite element method in the assumed displacement fields and the formulation of the variational functional. In contrast to the conventional method (based on the Rayleigh-Ritz mathematical technique) the Trefftz method (based on the Trefftz mathematical technique) assumes the displacement field is composed of two independent components; the intra-element displacement field which satisfies the governing differential equation and is used to approximate the variation of potential within the element domain, and the conforming frame field which specifically satisfies the inter-element continuity condition, defined on the boundary of the element. The frame field here is the same as that used in the conventional finite element method but defined strictly on the boundary of the element – hence the use of the term “hybrid” in the method’s nomenclature. The variational functional must thus include additional terms to account for boundary conditions, since the assumed solution field only satisfies the governing differential equation [1, 3].
, Kirchhoff plates, thick plates, general three-dimensional solid mechanics, antisymmetric solid mechanics, potential problems, shells, elastodynamic problems, geometrically nonlinear plate bending, and transient heat conduction analysis among various others [1, 3]. It is currently being applied to steady, non-turbulent, incompressible, Newtonian fluid
flow applications through ongoing research at the Faculty of Engineering and Information Technology (FEIT) at the Australian National University
(ANU) in Canberra, Australia. The hybrid Trefftz method is also being applied to some fields, e.g. computational modeling of hydrated soft tissues or water-saturated porous media, through ongoing research project at the Technical University of Lisbon
, Instituto Superior Técnico
in Portugal.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Trefftz method is a method for the numerical solution
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s named after the German
Germans
The Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematician Erich Trefftz (1888-1937). It falls within the class of finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
s.
Introduction
The hybrid Trefftz finite element method has been considerably advanced since its introduction about 30 years ago [1]. The conventional method of finite element analysis involves converting the differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
that governs the problem into a variational functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
from which element nodal properties – known as field variables – can be found. This can be solved by substituting in approximate solutions to the differential equation and generating the finite element stiffness matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
which is combined with all the elements in the continuum to obtain the global stiffness matrix [2]. Application of the relevant boundary conditions to this global matrix, and the subsequent solution of the field variables rounds off the mathematical process, following which numerical computations can be used to solve real life engineering problems [1, 3].
An important aspect of solving the functional requires us to find solutions which satisfy the given boundary conditions and satisfy inter-element continuity
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
since we define independently the properties over each element domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
[1].
The hybrid Trefftz method differs from the conventional finite element method in the assumed displacement fields and the formulation of the variational functional. In contrast to the conventional method (based on the Rayleigh-Ritz mathematical technique) the Trefftz method (based on the Trefftz mathematical technique) assumes the displacement field is composed of two independent components; the intra-element displacement field which satisfies the governing differential equation and is used to approximate the variation of potential within the element domain, and the conforming frame field which specifically satisfies the inter-element continuity condition, defined on the boundary of the element. The frame field here is the same as that used in the conventional finite element method but defined strictly on the boundary of the element – hence the use of the term “hybrid” in the method’s nomenclature. The variational functional must thus include additional terms to account for boundary conditions, since the assumed solution field only satisfies the governing differential equation [1, 3].
Advantages over conventional finite element method
The main advantages of the hybrid Trefftz method over the conventional method are:- the formulation calls for integrationIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
along the element boundaries only which allows for curve-sided or polynomial shapes to be used for the element boundary, - presents expansion bases for elements that do not satisfy inter-element continuity through the variational functional, and
- this method allows for the development of crack singular or perforated elements through the use of localized solution functions as the trial functions [1, 3].
Applications
Since its mainstream introduction some 30 years ago, this modified finite element method has become increasingly popular to applications such as elasticityElasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
, Kirchhoff plates, thick plates, general three-dimensional solid mechanics, antisymmetric solid mechanics, potential problems, shells, elastodynamic problems, geometrically nonlinear plate bending, and transient heat conduction analysis among various others [1, 3]. It is currently being applied to steady, non-turbulent, incompressible, Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
flow applications through ongoing research at the Faculty of Engineering and Information Technology (FEIT) at the Australian National University
Australian National University
The Australian National University is a teaching and research university located in the Australian capital, Canberra.As of 2009, the ANU employs 3,945 administrative staff who teach approximately 10,000 undergraduates, and 7,500 postgraduate students...
(ANU) in Canberra, Australia. The hybrid Trefftz method is also being applied to some fields, e.g. computational modeling of hydrated soft tissues or water-saturated porous media, through ongoing research project at the Technical University of Lisbon
Technical University of Lisbon
The Technical University of Lisbon is a Portuguese public university. It was created in 1930 in Lisbon, as a confederation of older schools, and comprises, nowadays, the faculties and institutes of veterinary medicine; agricultural sciences; economics and business administration; engineering,...
, Instituto Superior Técnico
Instituto Superior Técnico
Instituto Superior Técnico is a reputed school of engineering, part of Universidade Técnica de Lisboa . IST is the largest and the most prestigious school of engineering in Portugal...
in Portugal.