Structural FEA - understanding the fundamentals

[blue24]

TL;DR Summary

Are partial differential equations involved in transient, structural FEA?

I am a mechanical an engineer with a few years of experience. Most of the work I do is transient, structural finite element analysis. I have gotten reasonably competent at building models and pumping out results, but I regularly come across gaps in my fundamental knowledge. I have been doing some reading on the basics of finite element analysis because I want to understand more of its mathematical foundations.
My understanding is that the finite element method is often (primarily?) used to solve partial differential equations, which get pretty hairy. In structural analysis, the fundamental equation I am solving is the Equation of Motion:

$$F=m\ddot{x}+c\dot{x}+kx$$

This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis?

 

[FEAnalyst]

Actually FEA programs don't solve PDEs directly which may be counterintuitive. Instead they calculate so called stiffness matrix using the formula derived from weak form of PDE (in solid mechanics Navier's equation of motion). That's why when people write their own simple FEA programs (not only for 1D elements but also plane stress/strain) they don't have to implement PDEs. Dynamic analysis solves the following matrix formula: $M \ddot{u} + C \dot{u} +Ku=F$. In static analysis this reduces to $Ku=F$. To find out how FEA works it's best to understand some simple examples of hand calculations for beam and plane stress elements.

 

[Henryk]

Summary: Are partial differential equations involved in transient, structural FEA?

This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis

you are correct. This is an ordinary differential equation and in structural analysis PDEs are involved (partial derivatives with respect to x, y, z coordinates and, sometimes, time)

 

[blue24]

Thanks both of you. I know the strain displacement equations are PDE's. Is that where the PDE's come in? So in general, for a dynamic structural analysis,

Step 1 - Solve an ODE for displacements, based on applied boundary conditions and applied loading

Step 2 - Solve PDE's for strain from stress

I'm trying to put this in my own "layman's" terms, to make sure I understand. Thanks for the help!

 

[blue24]

Ok, so this question is still lingering in my mind - where do the PDE's come in for structural analysis? FEAnalyst says that FEA programs don't solve PDE's directly. I think I understand that part. But what is the PDE that is being "indirectly" solved? Is it the strain displacement equations?

 

[SCP]

In structural analysis, the fundamental equation I am solving is the Equation of Motion:

$$F=m\ddot{x}+c\dot{x}+kx$$
This is an ordinary differential equation, not a PDE. Are PDE’s involved in structural analysis?

This is only correct if the system you are studying has a single degree of freedom. For a lumped-parameter system with multiple degrees of freedom, this expands into the more general matrix version quoted by @FEAnalyst:

$M \ddot{u} + C \dot{u} +Ku=F$.

In the case of the more general distributed-parameter system, you get a PDE. However, computers don't like differential equations, and the numerical solution requires discretization. Discretization is what you are doing when you are meshing your system. The meshed (i.e. - discretized) version of the PDE returns us to the matrix equation above, and this is what your FEA program ultimately solves.

 

[blue24]

This is only correct if the system you are studying has a single degree of freedom. For a lumped-parameter system with multiple degrees of freedom, this expands into the more general matrix version quoted by @FEAnalyst:

In the case of the more general distributed-parameter system, you get a PDE. However, computers don't like differential equations, and the numerical solution requires discretization. Discretization is what you are doing when you are meshing your system. The meshed (i.e. - discretized) version of the PDE returns us to the matrix equation above, and this is what your FEA program ultimately solves.

Thank you, this is helpful!

 

[vco]

Ok, so this question is still lingering in my mind - where do the PDE's come in for structural analysis? FEAnalyst says that FEA programs don't solve PDE's directly. I think I understand that part. But what is the PDE that is being "indirectly" solved? Is it the strain displacement equations?

The underlying PDE is the Cauchy momentum equation.

It is just Newton II expressed for a continuum. Note that the $\mathbf f$ in it contains body forces only, like gravity. The more typical types of external forces come into play as boundary conditions for stress.

Edit: clarifications