# Finite Element Method Solved Problems Explicit analyses aim to solve for acceleration (or otherwise ).

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The finite element method (FEM) is a numerical problem-solving methodology commonly used across multiple engineering disciplines for numerous applications such as structural analysis, fluid flow, heat transfer, mass transport, and anything existing as a real-world force.

This practice systematically yields equations and attempts to approximate the values of the unknowns.

These events can be best exampled by extreme scenarios such as an automotive crash, ballistic event, or even meteor impact.

In these cases, the material models do not only need to account for the variation of stress with strain but also the strain rate.

For large models, inverting the matrix is highly expensive and will require advanced iterative solvers (over standard direct solvers).

Sometimes, this is also known as the backward Euler integration scheme.This method requires additional computation and can be harder to implement.However, it will be used in lieu of explicit methodologies when problems are still and using alternative analysis methods is impractical.For more information, this Wikipedia page provides great examples with illustrations of how both methodologies give numerical approximations to solutions of time-dependent and PDE equations.Explicit analysis offers a faster solution in events where there is a dynamic equilibrium or otherwise: The explicit method should be used when the strain rates/velocity is over 10 units/second or 10 m/s respectively.For all nonlinear and non-static analyses, incremental load (also known as displacement steps) are needed.In more simplistic terminology, this means we need to break down the physics/time relationship to solve a mathematical problem.Some interesting examples are also depicted in Figure 01.All of these problems are expressed through mathematical partial differential equations (PDE’s).While today’s computers can’t single-handedly solve PDE’s, they are equipped to solve matrix equations. In most structural problems, the nonlinear equations fall into 3 categories: In linear problems, the PDE’s reduce to a matrix equation as: [K] = and for non-linear static problems as: [K(x)] = For dynamic problems, the matrix equations come down to: [M] [C] [K] = where (.‘) represents the derivative.One method of solving for the unknowns is through matrix inversion (or equivalent processes). When the problem is nonlinear, the solution is obtained in a number of steps and the solution for the current step is based on the solution from the previous step.

## Comments Finite Element Method Solved Problems

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