Trefftz method

Key takeaways

Trefftz method belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Trefftz method to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Trefftz method from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz (1888–1937). It falls within the class of finite element methods.

Introduction The hybrid Trefftz finite-element method has been considerably advanced since its introduction by J. Jiroušek in the late 1970s. The conventional method of finite element analysis involves converting the differential equation 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. 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. An important aspect of solving the functional requires us to find solutions that satisfy the given boundary conditions and satisfy inter-element continuity since we define independently the properties over each element domain. 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.

Advantages over conventional finite element method The main advantages of the hybrid Trefftz method over the conventional method are:

the formulation calls for integration 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.

Applications This modified finite element method has become increasingly popular to applications such as elasticity, 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. 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.

Notes

References Qin, Q.H. (2000), The Trefftz Finite and Boundary Element Method, Southampton, England: WIT Press, pp. 1–55 Connor, J.J.; Brebbia, C.A. (1976), Finite Element Techniques for Fluid Flow (3rd ed.), Bristol, England: Newnes-Butterworths, Bibcode:1976butt.bookR....C Qin, Q.H. (2004), "Formulation of hybrid Trefftz finite element method for elastoplasticity", Applied Mathematical Modelling, 29 (2): 235–252, doi:10.1016/j.apm.2004.09.004

External links "Trefftz method", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Hybrid-Trefftz research project at Instituto Superior Técnico in Lisbon, Portugal

Worked examples

Example 1 — a first encounter with Trefftz method

Start with the simplest possible case. Write down what Trefftz method claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Trefftz method before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Trefftz method ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Trefftz method

In research: Trefftz method appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Trefftz method in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Trefftz method is common in secondary-school and first-year university syllabi. It links to neighbouring topics Numerical differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Trefftz method outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Trefftz method in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Trefftz method means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Trefftz method out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Trefftz method in simple terms?

In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz (1888–1937). It falls within the class of finite element methods.

Why does Trefftz method matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Trefftz method?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Trefftz method.