mathematics

Hu–Washizu principle

Hu–Washizu principle is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hu–Washizu principle rather than just read about it. In short: In continuum mechanics, and in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action ∫ V e [ 1 2 ε T C ε − σ T ε + σ T ( ∇ u ) − p ¯ T u ] d V − ∫ S σ e T ¯ T u d S {\displaystyle \int _{V^{e}}\left[{\frac {1}{2}}\varepsilon ^{T}C\varepsilon -\sigma ^{T}\varepsilon +\sigma ^{T}(\nabla u)-{\bar {p}}^{T}u\right]dV-\int _{S_{\sigma }^{e}}{\bar {T}}^{T}u\ d…

Key takeaways

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

Reference excerpt

In continuum mechanics, and in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action

∫ V e [ 1 2 ε T C ε − σ T ε + σ T ( ∇ u ) − p ¯ T u ] d V − ∫ S σ e T ¯ T u d S {\displaystyle \int _{V^{e}}\left[{\frac {1}{2}}\varepsilon ^{T}C\varepsilon -\sigma ^{T}\varepsilon +\sigma ^{T}(\nabla u)-{\bar {p}}^{T}u\right]dV-\int _{S_{\sigma }^{e}}{\bar {T}}^{T}u\ dS}

is stationary, where C {\displaystyle C} is the elastic stiffness tensor. The Hu–Washizu principle is used to develop mixed finite element methods. The principle is named after Hu Haichang and Kyūichirō Washizu. The Euler–Lagrange equations of the Hu–Washizu functional are the following equations:

{ ∇ ⋅ σ + f = 0 (Equilibrium) σ = ∂ W ∂ ε (Constitutive law) ε = ∇ s u (Compatibility) {\displaystyle {\begin{cases}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} =\mathbf {0} &{\text{(Equilibrium)}}\\{\boldsymbol {\sigma }}={\frac {\partial W}{\partial {\boldsymbol {\varepsilon }}}}&{\text{(Constitutive law)}}\\{\boldsymbol {\varepsilon }}=\nabla ^{s}\mathbf {u} &{\text{(Compatibility)}}\end{cases}}}

with appropriate boundary conditions

σ ⋅ n = t on ∂ Ω t {\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} =\mathbf {t} \quad {\text{on}}\quad \partial \Omega _{t}} .

References

Further reading K. Washizu: Variational Methods in Elasticity & Plasticity, Pergamon Press, New York, 3rd edition (1982) O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu : The Finite Element Method: Its Basis and Fundamentals, Butterworth–Heinemann, (2005).

Worked examples

Example 1 — a first encounter with Hu–Washizu principle

Start with the simplest possible case. Write down what Hu–Washizu principle 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 Hu–Washizu principle 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 Hu–Washizu principle 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 Hu–Washizu principle

In research: Hu–Washizu principle 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 Hu–Washizu principle 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: Hu–Washizu principle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Applied mathematics stubs, Calculus of variations, Continuum mechanics, so understanding it makes those chapters shorter.
In everyday life: Look for Hu–Washizu principle 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.

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How to study Hu–Washizu principle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hu–Washizu principle 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 Hu–Washizu principle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hu–Washizu principle in simple terms?

In continuum mechanics, and in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action ∫ V e [ 1 2 ε T C ε − σ T ε + σ T ( ∇ u ) − p ¯ T u ] d V − ∫ S σ e T ¯ T u d S {\displaystyle \int _{V^{e}}\left[{\frac {1}{2}}\varepsilon ^{T}C\vare…

Why does Hu–Washizu principle 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 Hu–Washizu principle?

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 Hu–Washizu principle.