Computational Models for the Human Body: Special Volume, by Nicholas Philippe Ayache

By Nicholas Philippe Ayache

Provides a greater figuring out of the physiological and mechanical behaviour of the human physique and the layout of instruments for his or her reasonable numerical simulations, together with concrete examples of such computational versions. This ebook covers a wide range of equipment and an illustrative set of applications.

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Additional info for Computational Models for the Human Body: Special Volume, Volume 12 (Handbook of Numerical Analysis)

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We have exploited the identity νD(u) : ∇v = Ω νD(u) : D(v), Ω which derives from the symmetry of the tensor D(u). S ECTION 11 39 The boundary term may now be split into two parts, v · 2νD(u) · n − pn = ∂Ω Γd v · 2νD(u) · n − pn + Γn v · h. We note that the contribution from the Neumann boundary is now a given data, while contribution from the Dirichlet boundary can be eliminated by appropriately choosing the test space V. By inspection, we may recognise that all terms make sense if we choose as test function spaces V = v ∈ H1 (Ω): v|Γ d = 0 , Q = q ∈ L2 (Ω): with q = 0 if Γ d = ∂Ω , Ω and if we seek, at each time t, the velocity in Vg = u ∈ H1 (Ω): u|Γ d = g and the pressure in Q.

N. 1 (Poincaré inequality – multidimensional case). Let f : RN → R be a function of H1 (Ω), with f = 0 on Γ ⊂ ∂Ω of strictly positive measure. Then there exists a positive constant CP (depending only on the domain Ω and on Γ ), such that f CP ∇f L2 (Ω) L2 (Ω) . 2. Let Ω be a bounded and connected subset of RN , where N = 2 or 3. Furthermore, let us assume that the velocity field u ∈ H1 (Ω) vanishes on Γ ⊂ ∂Ω of strictly positive measure. Then, there exists a constant CK > 0 so that the following inequality holds: D(u) : D(u) CK ∇u Ω 2 .

Let Vt ⊂ Ωt be a subdomain of Ωt and let us consider the function f : I × Vt → R. Then f is integrable on Vt if and only if (f ◦ Lt )Jt is integrable on V0 = L−1 t (Vt ), and fˆ(t, ξ ) Jt (ξ ) dξ , f (t, x) dx = Vt V0 where fˆ(t, ξ ) = f (t, Lt (ξ )). In short, fˆJt . 5. The Reynolds transport theorem An interesting property of the Jacobian is that its time derivative is linked to the divergence of the velocity field. 1. 8) in the Eulerian frame. Then D Jt = Jt div u. Dt This relation is sometimes called Euler expansion formula.

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