By Nicholas Philippe Ayache
Offers 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 functions.
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Extra info for Computational Models for the Human Body: Special Volume XII (Handbook of Numerical Analysis)
Indeed, 1 ∇ |u|2 · u Ω Ω 2 1 1 =− |u|2 div u + |u|2 u · n. 2 Ω 2 ∂Ω c(u, u, u) = (u·∇)u · u = Now, the last integral is zero since u = 0 on ∂Ω. Moreover, for the same reason div u = Ω u · n = 0. ∂Ω Then, if we set c= |u|2 , Ω we have |u|2 div u = Ω |u|2 div u − c Ω |u|2 − c div u div u = Ω = b u, |u|2 − c Ω = 0, where the last equality is obtained since (|u|2 − c) ∈ Q and b(u, q) = 0, ∀q ∈ Q. 1. P ROOF. 11). We have 1 d u 2 dt Then, 2 L2 (Ω) + c(u, u, u) + b(u, p) + a(u, u) = (f, u). 1) 1 d u 2L2 (Ω) + a(u, u) = (f, u).
1 of G IRAULT and R AVIART . The application L defined as L(v) = ∂u , v + a(u, v) + c(u, u, v) − (f, v), ∂t ∀v ∈ V, belongs to V , being a linear continuous functional on V. 2 and obtain the desired result. 12. 8), by which we may assess a continuous dependence of the solution from the given data. 1 (Energy inequalities). 8), ∀t ∈ I . Then the following inequalities hold: u(t) 2 L2 (Ω) t + C1 ∇u(τ ) 0 2 dτ L2 (Ω) u0 2 L2 (Ω) t + f(τ ) 0 2 dτ L2 (Ω) et , where C1 = 4ν0 CK , and u(t) 2 L2 (Ω) t + C2 0 ∇u(τ ) 2 dτ L2 (Ω) u0 2 L2 (Ω) + CP C2 t f(τ ) 0 2 dτ, L2 (Ω) where C2 = 2ν0 CK .
29c) 54 A. Quarteroni and L. Formaggia C HAPTER III This algorithm is known as algebraic Chorin–Temam scheme. 19) with a “discrete Laplacian” DM −1 D T , which incorporates the boundary condition of the original problem. No additional boundary condition is required for the pressure, contrary to the standard (differential type) scheme. 1. The finite element mass matrix M is sparse and with the same graph structure as C. 24). However, the matrix M may be approximated by a diagonal matrix called lumped mass matrix (Q UARTERONI and VALLI ), whose inversion is now trivial.