Continuum mechanics/Partial differential equations

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Partial differential equations (PDEs) are the most common method by which we model physical problems in engineering. Finite element methods are one of many ways of solving PDEs. This handout reviews the basics of PDEs and discusses some of the classes of PDEs in brief. The contents are based on Partial Differential Equations in Mechanics volumes 1 and 2 by A.P.S. Selvadurai and Nonlinear Finite Elements of Continua and Structures by T. Belytschko, W.K. Liu, and B. Moran.

Contents

[edit] Partial Differential Equations

A PDE is a relationship between an unknown function of several variables and its partial derivatives.

Let u(x1,x2,x3,t) be an unknown function. The independent variables are x1, x2, x3, and t. We usually write

u = u(x1,x2,x3,t)

and say that u is the dependent variable.

Partial derivatives are denoted by expressions such as

u_{,1} = \frac{\partial u}{\partial x_1} ~;~~ u_{,2} = \frac{\partial u}{\partial x_2} ~;~~ u_{,11} = \frac{\partial^2 u}{\partial x_1\partial x_1} \equiv \frac{\partial^2 u}{\partial x_1^2} ~;~~ u_{,12} = \frac{\partial^2 u}{\partial x_1\partial x_2}~.

Some examples of partial differential equations are

\begin{align} u_{,t} = u_{,1} + u_{,2} &\Leftrightarrow \frac{\partial u}{\partial t} = \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} \\ \nabla^2 u = 0 \Leftrightarrow u_{,11} + u_{,22} + u_{,33} = 0 &\Leftrightarrow \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = 0 \\ u_{,1111} = u_{,22} + u &\Leftrightarrow \frac{\partial^4 u}{\partial x_1^4} = \frac{\partial^2 u}{\partial x_2^2} + u ~. \end{align}

An example of a system of partial differential equations is

\boldsymbol{\nabla} (\boldsymbol{\nabla} \bullet \mathbf{u}) + \nabla^2 \mathbf{u} + \mathbf{f} = \mathbf{0} \Leftrightarrow u_{k,ki} + u_{i,jj} + f_i = 0

In expanded form this system of equations is

\begin{align} \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2\partial x_1} + \frac{\partial^2 u_3}{\partial x_3\partial x_1} + \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_1}{\partial x_2^2} + \frac{\partial^2 u_1}{\partial x_3^2} + f_1 & = 0 \\ \frac{\partial^2 u_1}{\partial x_1\partial x_2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3\partial x_2} + \frac{\partial^2 u_2}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_2}{\partial x_3^2} + f_2 & = 0 \\ \frac{\partial^2 u_1}{\partial x_1\partial x_3} + \frac{\partial^2 u_2}{\partial x_2\partial x_3} + \frac{\partial^2 u_3}{\partial x_3^2} + \frac{\partial^2 u_3}{\partial x_1^2} + \frac{\partial^2 u_3}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3^2} + f_3 & = 0 \end{align}

It is often more convenient to write PDEs in vector notation or index notation.

[edit] Order of a PDE

The order of a PDE is determined by the highest derivative in the equation. For example,

\begin{align} \frac{\partial u}{\partial x_1} - \frac{\partial u}{\partial x_2} & = 0 ~~~\text{is a first-order PDE.}\\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} & = 0 ~~~~\text{is a second-order PDE.}\\ \frac{\partial^4 u}{\partial x_1^4} + \frac{\partial^2 u}{\partial x_2^2} - u & = 0 ~~~~\text{is a fourth-order PDE.}\\ \left(\frac{\partial u}{\partial x_1}\right)^3 + \frac{\partial u}{\partial x_2} + u^4 & = 0 ~~~\text{is a first-order PDE.} \end{align}

[edit] Linear and nonlinear PDEs

A linear PDE is one that of the first degree of its field variable and partial derivatives. For example,

\begin{align} \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} & = 0 ~~~\text{is linear}~.\\ \frac{\partial u}{\partial x_1} + \left(\frac{\partial u}{\partial x_2}\right)^2 & = 0 ~~~\text{is nonlinear}~.\\ \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} + u^2 & = 0 ~~~\text{is nonlinear}~.\\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} & = x_1 ~~~\text{is linear}~.\\ \frac{\partial^2 u}{\partial x_1^2} + u\frac{\partial^2 u}{\partial x_2^2} & = 0 ~~~\text{is quasilinear}~. \end{align}

The above equations can also be written in operator notation as

\begin{align} D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2}~. \\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \left(\frac{\partial }{\partial x_2}\right)^2~.\\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2} + u^2~.\\ D(u) = x_1 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + \frac{\partial^2 }{\partial x_2^2}~.\\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + u\frac{\partial^2 }{\partial x_2^2}~. \end{align}

[edit] Homogeneous PDEs

Let L be a linear operator. Then an linear partial differential equation can be written in the form

L(u) = f(x_1,x_2,x_3,t)~.

If f(x1,x2,x3,t) = 0, the PDE is called homogeneous.

[edit] Elliptic, Hyperbolic, and Parabolic PDEs

We usually come across three-types of second-order PDEs in mechanics. These are classified as elliptic, hyperbolic, and parabolic.

The equations of elasticity (without inertial terms) are elliptic PDEs. Hyperbolic PDEs describe wave propagation phenomena. The heat conduction equation is an example of a parabolic PDE.

Each type of PDE has certain characteristics that help determine if a particular finite element approach is appropriate to the problem being described by the PDE. Interestingly, just knowing the type of PDE can give us insight into how smooth the solution is, how fast information propagates, and the effect of initial and boundary conditions.

  • In hyperbolic PDEs, the smoothness of the solution depends on the smoothness of the initial and boundary conditions. For instance, if there is a jump in the data at the start or at the boundaries, then the jump will propagate as a shock in the solution. If, in addition, the PDE is nonlinear, then shocks may develop even though the initial conditions and the boundary conditions are smooth. In a system modeled with a hyperbolic PDE information travels at a finite speed called the wavespeed. Information is not transmitted until the wave arrives.
  • In contrast, the solutions of elliptic PDEs are always smooth, even if the initial and boundary conditions are rough (though there may be singularities at sharp corners). In addition, boundary data at any point affect the solution at all points in the domain.
  • Parabolic PDEs are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.

Suppose we have a second-order PDE of the form

a(x_1,x_2) \frac{\partial^2 u}{\partial x_1^2} + b(x_1,x_2) \frac{\partial^2 u}{\partial x_1\partial x_2} + c(x_1,x_2) \frac{\partial^2 u}{\partial x_2^2} + d(x_1,x_2) \frac{\partial u}{\partial x_1} + e(x_1,x_2) \frac{\partial u}{\partial x_2} + f(x_1,x_2) u = g(x_1,x_2)

Then, the PDE is called elliptic if

{ b^2 - 4ac < 0 ~~~~\implies~~~~ \text{ elliptic} ~. }

An example is

\frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}

The PDE is called hyperbolic if

{ b^2 - 4ac > 0 ~~~~\implies~~~~ \text{ hyperbolic} ~. }

An example is

\frac{\partial^2 u}{\partial x_1^2} + 3\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}


The PDE is called parabolic if

{ b^2 - 4ac = 0 ~~~~\implies~~~~ \text{ parabolic} ~. }

An example is

\frac{\partial^2 u}{\partial x_1^2} + 2\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1}

[edit] Important PDEs in mechanics

  • Laplace's equation.
\nabla^2 u = 0 \equiv \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = 0 ~.

Appears in almost every field of physics.

  • Poisson's equation.
\nabla^2 u = -f \equiv \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = - f ~.

Appears is almost every field of physics.

  • Heat/Diffusion equation.
\alpha\nabla^2 T = \frac{\partial T}{\partial t}~.
  • Wave equation.
\frac{\partial^2 u}{\partial x_1^2} - \cfrac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = -f(x,t)~.
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