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Solving 3-D Problems Using 2-D Models

Partial Differential Equation Toolbox™ software solves problems in two space dimensions and time, whereas reality has three space dimensions. The reduction to 2-D is possible when variations in the third space dimension (taken to be z) can be accounted for in the 2-D equation. In some cases, like the plane stress analysis, the material parameters must be modified in the process of dimensionality reduction.

When the problem is such that variation with z is negligible, all z-derivatives drop out and the 2-D equation has exactly the same units and coefficients as in 3-D.

Slab geometries are treated by integration through the thickness. The result is a 2-D equation for the z-averaged solution with the thickness, say D(x,y), multiplied onto all the PDE coefficients, c, a, d, and f, etc. For instance, if you want to compute the stresses in a sheet welded together from plates of different thickness, multiply Young's modulus E, volume forces, and specified surface tractions by D(x,y), Similar definitions of the equation coefficients are called for in other slab geometry examples and application modes.

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