## Software Analysis

An important part of the design process is the simulation of the performance of a designed device. As an example a fastener or snap-fit is designed to work under certain static or dynamic loads (Chapter 4). The temperature distribution in an electronic chip may need to be calculated to determine the heat transfer behavior and possible thermal stress. Turbulent flow over a turbine blade controls cooling but may induce vibration. Whatever the device being designed, there are many possible influences on the device's performance.

These types of loads can be calculated using FEA. The analysis divides a given domain into smaller, discrete fundamental parts called elements. An analysis of each element is then conducted using the required mathematics. Finally, the solution to the problem as a whole is determined through an aggregation of the individual solutions of the elements. In this manner, dividing the problem into smaller and simpler problems upon which approximate soludons can be obtained can solve complex problems. General-purpose FEA software programs have been generalized such that users do not need to have detailed knowledge of FEA. A FEA model can be thought of as a system of solid blocks (elements) assembled together. Several types of elements that are available in the finite-element library are FEA packages such as NASTRAN and ANSYS.

When a structure is modeled, individual sets of matrix equations are automatically generated for each element. The elements in the model share common nodes so that individual sets of matrix equations can be combined into a set of matrix equations. This set relates all of the nodal deflections to the nodal forces. Nodal deflections are solved simultaneously from the matrix. When displacements for all nodes are known, the state of deformation of each element is known and stress can be determined through stress-strain relations.

With a 2-D structure problem, each node displacement has three degrees of freedom, one translational in each of x and y directions and a rotational in the (x-y) plane. In a 3-D structure problem, the displacement vector can have up to six degrees of freedom for each nodal point. Each degree of freedom at a nodal point may be unconstrained (unknown) or constrained. The nodal constraint can be given as a fixed value or a defined relation with its adjacent nodes. One or more constraints must be given prior to solving a structure problem.

FEA obtains stresses, temperatures, velocity potentials, and other desired unknown variables in the analyzed model by minimizing an energy function. The law of conservation of energy is a well-known principle of physics. It states that, unless atomic energy is involved, the total energy of a system must be zero. Thus, the finite element energy functional must equal zero. The finite element method obtains the correct solution for any analyzed model by minimizing the energy functional. Thus, the obtained solution satisfies the law of conservation of energy.

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