Motivation

• Scope
  • Mathematical models of minimisation of a potential (energy)
  • Involving partial differential equations (PDEs)
• Challenges
  • Often impossible to find an analytical solution
  • Infinite-dimensional, cannot be mapped to computers
• Solution
  • Approximate finite-dimensional models (FEM, FDM, FVM, ...)

Motivation

Naturally,

approximations make sense on well-posed minimisation problems.

In this talk...

1. How do we study mathematical models of minimisation problems?
2. How do we approximate them with finite elements?

Contents

• Motivation
• Mathematical theory of potential energy minimisation
• The finite element method (FEM)
• Gridap FEM tutorial

Mathematical models

• Minimisation problem
• Variational formulation
• Differential equation
• Abstract theory

Minimisation problem

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• $u(0), u(1) \equiv 0$ 🠒 boundary conditions (clamped)
• $f(x)$ 🠒 forcing term
• $k(x) \equiv 1$ 🠒 Young's modulus

Minimisation problem

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• Assume
  • Material in the elastic regime
  • Small displacements $(u(x) \llless 1)$

Minimisation problem

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• Assume
  • Material in the elastic regime
  • Small displacements $(u(x) \llless 1)$

Question 1.

• What physical law expresses the elastic energy stored in the bar?

Minimisation problem

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• Hooke's law: $\qquad E(u) = \frac{1}{2} \int_0^1 u'(x)^2 \ \mathrm{d}x$
• Total potential: $\quad J(u) \doteq \int_0^1 \left( \frac{1}{2} u'(x)^2 - f(x)u(x) \right) \ \mathrm{d}x$
• Let $\mathcal{C}_0^1([0,1]) = \left\{ v \in \mathcal{C}^1([0,1]) : v(0) = v(1) = 0 \right\}$
• Minimum potential energy principle:

Minimisation problem

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• Same minimisation problem for, e.g.,
  • a fixed elastic cord under transversal load
  • heat conduction on a bar
• Formulated in an infinite dimensional space of functions

Variational formulation

• Seek an equivalent formulation using calculus of variations
  • Minimisation problem 🠒 Variational formulation
  • Mathematical advantages (to be stated soon)

Variational formulation

• Assume there exists a global minimum $u$ for $\text{\textsf{\color{brown}(M)}}$
• Given $v \in \mathcal{C}_0^1([0,1])$ and $\alpha \in \mathbb{R}$, let

• Since $u$ is a global minimiser,

Variational formulation

• $\Phi_v (\alpha)$ has a global minimum at zero
• If $\Phi_v (\alpha)$ is differentiable, then

Variational formulation

• Algebraic manipulations + cancelling high-order terms gives

• The weak or variational form of the physical problem $\text{\textsf{\color{brown}(M)}}$
• In statics, referred to as principle of virtual work:
  • Any perturbation of the equilibrium configuration requires to supply energy to the system $(J(u) \leq J(u + \alpha v))$

Differential equation

• Now, relate $\text{\textsf{\color{brown}(M)}}$ and $\text{\textsf{\color{orange}(V)}}$ with standard differential equations
• Assume $u \in \mathcal{C}^2([0,1])$ (and also $f \in \mathcal{C}^0([0,1])$)
• Using extra regularity of $u$, integrate $\text{\textsf{\color{orange}(V)}}$ by parts

Differential equation

• The fundamental lemma of calculus of variations leads to

• Thus,

Remark. Point-wise vs weak sense

But is the problem well-posed?

• $\text{\textsf{\color{brown}(M)}}$, $\text{\textsf{\color{orange}(V)}}$ and $\text{\textsf{\color{pink}(D)}}$: equivalent statements of same $\infty$-dim problem
• Note that $\text{\textsf{\color{pink}(D)}}$ assumes higher regularity (classical solutions)
• Proving existence and uniqueness of $\text{\textsf{\color{pink}(D)}}$ solutions is often hard
• In contrast, it is much easier to prove it for $\text{\textsf{\color{orange}(V)}}$ solutions
• Remark. Many problems do not have a solution in classical sense,
  e.g. clamped elastic L-shaped plate

Abstract setting

• Goal:
  • State some general abstract existence and uniqueness results
• Assume minimisation problems, related to an energy functional $J$, on a space of functions defined in a bounded $\Omega \subset \mathbb{R}^d$, $d = 2,3$.

Abstract setting

Definition. (Bi)linear forms. Given a vector space $V$ in the field of scalars $\mathbb{R}$, a form $\ell : V \to \mathbb{R}$ is linear if it satisfies:

A (two-argument) form $a : V \times V \to \mathbb{R}$ is a bilinear form if it is linear with respect to each of its two arguments separately.

Abstract setting

Definition. Quadratic minimisation problem. Let us consider a functional $J : V \to \mathbb{R}$ on a vector space $V$ expressed as

for a symmetric bilinear form $a : V \times V \to \mathbb{R}$, a linear form $\ell : V \to \mathbb{R}$ and $c \in \mathbb{R}$. The quadratic minimisation problem on $V$ is

Abstract setting

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

• $a(u,v) = \int_0^1 u'(x)v'(x), \enskip \forall u,v \in \mathcal{C}_0^1([0,1])$
• $\ell(v) = \int_0^1 f(x)v(x), \enskip \forall v \in \mathcal{C}_0^1([0,1])$
• $c = 0$

Abstract setting

Perturbing (again) around the global minimiser $u \in V$ with $v \in V$ and $\alpha \in \mathbb{R}$,

we obtain a linear variational problem.

Abstract setting

Definition. A linear variational problem reads as

where $V$ is a vector space, $a : V \times V \to \mathbb{R}$ is a bilinear form and $\ell : V \to \mathbb{R}$ is a linear form.

Remark. More generally, $u \in \tilde{V}$, where $V$ is an affine space.

Abstract setting

Definition. A symmetric bilinear form $a : V \times V \to \mathbb{R}$ on a real vector space $V$ is positive definite if

Definition. A symmetric positive definite bilinear form $a : V \times V \to \mathbb{R}$ induces the energy norm

Abstract setting

Definition. Continuity of (bi)linear forms. Given a vector space $V$ with norm $\| \cdot \|$, a linear form $\ell : V \to \mathbb{R}$ is continuous or bounded if

A bilinear form $a : V \times V \to \mathbb{R}$ is continuous if

Abstract setting

Theorem. Existence and uniqueness of a minimiser in finite dimension. If the space $V$ in $\text{\textsf{\color{brown}(M)}}$ has finite dimension and involves a positive definite symmetric bilinear form $a$ and a continuous functional $\ell$, there exists a unique solution for this problem.

Remark. Under finite dimensionality, the variational form can be recast into a solvable square linear system.

Functional spaces

In infinite dimensions:

• Choice of functional space $V$ for solution of $\text{\textsf{\color{brown}(M)}}$ is critical
• In general, we seek the largest $V$ for which
  • $a$ has sense (i.e., is bounded) and
  • $V$ satisfies suitable boundary conditions

Functional spaces

In infinite dimensions:

• If $V$ is too small, there will not exist a solution to $\text{\textsf{\color{brown}(M)}}$
• Example 2. There is no solution to $J \doteq \frac{1}{2} \int_0^1 u^2(x) - u(x) \mathrm{d}x$ in $\mathcal{C}_0^0([0,1])$ (no continuous fun. minimises $\text{\textsf{\color{brown}(M)}}$ and satisfies BCs)
• The key requirement for $V$ is completeness and, in particular, $V$ is generally a Hilbert space (normed, complete + with inner product)

Functional spaces

Theorem. Existence and uniqueness of solutions in Hilbert spaces. Let us consider a Hilbert space $V$ endowed with the inner product $a : V \times V \to \mathbb{R}$ and a linear functional $\ell : V \to \mathbb{R}$. The quadratic minimisation problem

has a unique solution.

Functional spaces

Definition. The $L^2(\Omega)$ space is the space of square-integrable functions on $\Omega$:

It is endowed with the inner product

Functional spaces

Definition. The $H^1(\Omega)$ space is the space of square-integrable functions with square integrable gradients on $\Omega$:

It is endowed with the inner product

Functional spaces

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

The previous theorem holds for $V = H_0^1([0,1])$.

Functional spaces

Example 3. A multidimensional version of Example 1. in $\Omega \subset \mathbb{R}^d$.

Models, e.g., the normal displacement $u$ of a membrane under a normal external pressure $f$ and prescribed displacement at $\partial \Omega$.

Assuming clamped, the previous theorem holds for $V = H_0^1(\Omega)$.

Boundary value problem

Proposition. Green's first formula (integration by parts). For all $\boldsymbol{\psi} \in \mathcal{C}^1 (\overline{\Omega}), v \in \mathcal{C}^1 (\overline{\Omega})$, it holds:

Applying the proposition $(u \in \mathcal{C}^2 (\overline{\Omega}))$ and the fundamental lemma of calculus of variations to the variational form $\text{\textsf{\color{orange}(V)}}$ of Example 3....

The finite element method

• Our goal now is to approximate the mathematical model
• Map an $\infty$-dimensional $\text{\textsf{\color{brown}(M)}}$ to a numerical approximation
• FEM approximation derived from variational formulation $\text{\textsf{\color{orange}(V)}}$

The finite element method

• The Galerkin method
• Finite elements
• Error analysis

The Galerkin method

Definition. Galerkin approximation. Given the abstract variational formulation

let us consider a finite dimensional vector subspace $V_N \subset V$, with $\mathrm{dim}(V_N) = N$. The Galerkin approximation of $\text{\textsf{\color{orange}(V)}}$ reads

The Galerkin method

Since $V_N$ is a real vector space of dim $< \infty$, we can pick a basis:

Definition. Basis of a finite dimensional vector space. Given a finite dimensional real vector space $V_M$, the set $\{\varphi_1,\ldots,\varphi_M\} \subset M \in \mathbb{N}$ is a basis of $V_M$ if, for all $v \in V_M$, there is a unique set of coefficients $\{ v_i \}_{i=1}^M \subset \mathbb{R}$ such that $v = v_1 \varphi_1 + \ldots + v_M \varphi_M$. $M$ is equal to the dimension of $V_M$.

The Galerkin method

Lemma. Galerkin system as a linear system. We consider a basis $\mathcal{B} = \{ \varphi_1, \ldots, \varphi_N \}$ of $V_N$. The Galerkin problem $\text{\textsf{\color{yellow}(G)}}$ is equivalent to the linear system:

Find $u_N = \sum_{i=1}^N \mu_i \varphi_i$ with

Finite elements

• Choice of basis dramatically affects how "easy" and "accurate" we can solve $\text{\textsf{\color{lime}(S)}}$ in a finite precision machine
• Finite elements offer good compromise between well-conditioning, memory requirements, ease of implementation...
• Grounded on two ingredients
  • A partition (or mesh) of $\Omega$ into elements, aka, cells
  • Globally continuous functions that are polynomials in the cells

Finite elements

Example 4. A mesh generated with GMSH. From GridapGMSH.

Finite elements

Definition. 1D mesh. Given a domain $\Omega \doteq [a,b] \subset \mathbb{R}$, $M \in \mathbb{N}$, and a set of nodes

we can define a mesh $\mathcal{M}_M$ of $\Omega$ as

The $(x_{j-1},x_j)$ are the cells of $\mathcal{M}_M$, w. cell size $h_j \doteq \left| x_j - x_{j-1} \right|$.

Finite elements

Definition. Hat function in 1D linear finite elements. Given a mesh $\mathcal{M}_{N+1}$, for every interior node $x_j$, $j = 1, \ldots, N$, we define its hat function $\varphi_j$ as

Finite elements

Definition. Hat function in 1D linear finite elements (continues). At the end-points, i.e., $j \in \{ 0, N + 1 \}$, the hat functions are

Finite elements

Finite elements

Example 1. Straight elastic bar under tangential load in $\Omega = [0,1]$

We consider $f \equiv 1$ and a uniform mesh with $N+1$ cells of size $h$ (uniform = equidistant nodes). We want to compute the system $\text{\textsf{\color{lime}(S)}}$ with linear finite elements.

Finite elements

Using previous Lemma, the Galerkin approximation $\text{\textsf{\color{yellow}(G)}}$ leads to the linear system $\mathbf{A} \boldsymbol{\mu} = \mathbf{f}$ $\text{\textsf{\color{lime}(S)}}$ with

We generally use a numerical quadrature to integrate $\mathbf{A}_{ij}$ and $\mathbf{f}_{j}$. Assuming a trapezoidal rule to integrate $\mathbf{f}_{j}$, the linear system reads:

Finite elements

Finite elements

• Since (homogeneous) boundary conditions, we remove the equations for $\mu_0$ and $\mu_{N+1}$.
• The linear system is tridiagonal. In general, FEs yield sparse matrices (reduces memory storage and can use of sparse solvers).
• Questions.
  • What is the approximation error?
  • How does it depend on $N$?

Finite elements

Proposition. Error estimates for the FE solution. Assuming

• a well-posed $\text{\textsf{\color{yellow}(G)}}$ in $\Omega \subset \mathbb{R}^d$ with a FE space $V_N$ of order $p$,
• exact imposition of the boundary conditions, and
• $u \in H^{q+1}(\Omega)$ for $q \geq p$,

then, we have the a priori error estimates

Gridap FEM tutorials

• Gridap 🠒 Software for FEM 100% in Julia (Why in Julia?)
• Well maintained Gridap tutorials

Conclusions

• FEM approximates PDEs related to energy minimisation problems
• Variational formulation are the best for mathematical analysis
• Choosing the functional spaces is essential for well-posedness
• FEM is an $N$-dim approximation of an $\infty$-dim variational problem
• FEM rely on meshes and cellwise polynomial bases
  • Admits very general geometry and BCs
  • Sparse "well-conditioned" linear systems

References

Material from this lecture adapted from the lecture notes by Prof. Santiago Badia at Monash University, Melbourne, Australia.

1. C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Publications, 2009.

2. O. C. Zienkiewicz, R. L. Taylor, J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, 2013.

And that's all! Thank you!