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# 12.4. Using Excel to Help Solve Problems Formulated Using the Finite Element Method

## Problem

You've applied the finite element method to a nonlinear problem and must now iteratively find a solution.

## Solution

Use Solver to find a solution to the resulting nonlinear equation or system of equations. See Chapter 9 for recipes showing how to use Solver.

## Discussion

The finite element method is a very popular method for numerically solving differential equations. The method involves discretizing the problem domain and governing equations, resulting in an algebraic system of equations that must be solved to arrive at an approximate solution to the problem. In some cases, the problem is nonlinear, which results in a nonlinear equation, or system of equations, that must be solved. As discussed in Chapter 9, Excel's built-in Solver tool is very adept at finding solutions to such equations.

By way of example, consider a heat conduction problem presented by Kythe and Wei in their Introduction to Linear and Nonlinear Finite Element Analysis.[*] The governing partial differential equation, with boundary conditions, is as follows:

This equation represents steady state heat conduction in a laterally insulated rod with constant temperature maintained at one end (x = 0) and with radiation heat transfer at the other end (x = L). k is the thermal conductivity of the rod material. σ is the Stefan-Boltzman constant. ...

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