CAE-Companion-2018-2019

Theory WISSEN CAE

element of the FE mesh while the Kernel of the central particle has been “sketched” as spanning its neighbors in the SPHmesh.

Observe that these equations are different from those we habitually solve in the FE method. Moreover, starting from the same equation but following a different set of algebraic operations and approximations we could arrive at the follow- ing alternative momentum equations :

Figure 1: Comparison between FE and SPH modeling of the same patch of 9 elements The above mentioned approximations can be applied to solve the continuity equations by parts whereby the partial derivatives of the unknowns are replaced by derivatives of the (known) kernels, yielding :

Artificial viscosity to handle shock discontinuities is required, just as for existing numerical integration methods. Note that the above set of resulting equations is not unique, but it has been found that the differences are usually small.

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