Drucker-Prager Elastoplasticity for Sand Animation: Supplementary Technical Document
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چکیده
The tensor M has 3 = 81 entries and no symmetries. Both the construction and application of M are somewhat expensive, and both can be avoided. If F = UΣV , then it turns out that Z(F, α) = UẐ(Σ, α)V and W(F) = UŴ(Σ)V , where Ẑ(Σ, α) and Ŵ(Σ) are diagonal matrices. It follows then that Y(F) = UŶ(Σ)V , with Ŷ(Σ) = Ŵ(Ẑ(Σ, α)), where Ŷ(Σ) is also a diagonal matrix. To be able to carry out these steps, it is required of the energy density function ψ, that it depends only on the singular values of F. In essence, we need to be able to define ψ̂ such that, ψ̂(Σ) = ψ(F). This allows us to write the definition of Ŵ as Ŵ(Σ) = ∂ψ̂ ∂Σ (Σ). Note that we have taken advantage of these relationships to avoid computing the singular value decomposition more often than necessary. Indeed, Ŵ is implemented by Energy derivative, and Ẑ is implemented by Project. Since these functions are rather simple in diagonal space, it might not be too surprising that the derivatives are also simpler there. Let M̂ be the diagonal space version of M, defined by
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Drucker-Prager Elastoplasticity for Sand Animation: Supplementary Technical Document
The tensor M has 3 = 81 entries and no symmetries. Both the construction and application of M are somewhat expensive, and both can be avoided. If F = UΣV , then it turns out that Z(F, α) = UẐ(Σ, α)V and W(F) = UŴ(Σ)V , where Ẑ(Σ, α) and Ŵ(Σ) are diagonal matrices. It follows then that Y(F) = UŶ(Σ)V , with Ŷ(Σ) = Ŵ(Ẑ(Σ, α)), where Ŷ(Σ) is also a diagonal matrix. To be able to carry out these ste...
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