Download Computational Models for the Human Body: Special Volume XII by Nicholas Philippe Ayache PDF

By Nicholas Philippe Ayache

Presents a greater realizing of the physiological and mechanical behaviour of the human physique and the layout of instruments for his or her practical numerical simulations, together with concrete examples of such computational versions. This publication covers a wide variety of equipment and an illustrative set of functions.

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Extra resources for Computational Models for the Human Body: Special Volume XII (Handbook of Numerical Analysis)

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2. The material derivative We can relate time derivatives computed with respect to the different frames. The material (or Lagrangian) time derivative of a function f , which we will denote Df /Dt , is defined as the time derivative in the Lagrangian frame, yet expressed as function of the Eulerian variables. That is, if f : I × Ωt → R and fˆ = f ◦ Lt , Df ∂ fˆ Df : I × Ωt → R, (t, x) = (t, ξ ), Dt Dt ∂t Therefore, for any fixed ξ ∈ Ω0 we may also write ξ = L−1 t (x). 3) d Df (t, x) = f t, x(t, ξ ) , Dt dt by which we can observe that the material derivative represents the rate of variation of f along the trajectory Tξ .

Then we may formally compute the discrete pressure terms by − DC −1 D T Π = −DC −1 Fs . Proving that A is non-singular thus reduces to show that the matrix S = DC −1 D T is non-singular. If we take any q ∈ RNQh with |q| = 0 we have by hypothesis that D T q = 0. Then qT Sq = D T q C −1 D T q = 0, T 48 A. Quarteroni and L. Formaggia C HAPTER III since C −1 is symmetric positive definite. Thus, matrix S (which is clearly symmetric) has all eigenvalues different from zero and, consequently, is non-singular.

3)) = dt Dt We have now the following fundamental result. 2 (Reynolds transport theorem). Let V0 ⊂ Ω0 , and Vt ⊂ Ωt be its image under the mapping Lt . Let f : I × Ωt → R be a continuously differentiable function with respect to both variables x and t. Then, d dt f= Vt Vt Df + f div u = Dt Vt ∂f + div(f u) . 11) P ROOF. 3), we have d dt f (t, x) dx = Vt d dt fˆ(t, ξ )Jt (ξ ) dξ = V0 = V0 V0 ∂ ˆ f (t, ξ )Jt (ξ ) dξ ∂t ∂ fˆ ∂ (t, ξ )Jt (ξ ) + fˆ(t, ξ ) Jt (ξ ) dξ . 3) to write V0 ∂ fˆ (t, ξ )Jt (ξ ) dξ = ∂t Vt Df (t, x) dx.

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