By Nicholas Philippe Ayache
Offers a greater knowing of the physiological and mechanical behaviour of the human physique and the layout of instruments for his or her real looking numerical simulations, together with concrete examples of such computational types. This booklet covers a wide variety of tools and an illustrative set of purposes.
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Extra resources for Computational Models for the Human Body: Special Volume XII (Handbook of Numerical Analysis)
I=1 j =1 Analogously, a vector (or a tensor) function f belongs to Hm (Ω) if all its components belong to H m (Ω), and we have 1/2 N f Hm (Ω) = fi 2 H m (Ω) , i=1 while its semi-norm is 1/2 N |f|Hm (Ω) = |fi |2H m (Ω) . i=1 35 36 A. Quarteroni and L. Formaggia C HAPTER III It is understood that, when m = 0, H0 (Ω) ≡ L2 (Ω). When equipped with the following scalar product: N (f, g)Hm (Ω) = (fi , gi )H m (Ω) , f, g ∈ Hm (Ω), i=1 Hm (Ω) the space is a Hilbert space. To ease notation, we will often use the following short-hand notation for the L2 scalar products: (v, w) ≡ (v, w)L2 (Ω) , (p, q) ≡ (p, q)L2 (Ω) .
24a) D Π = −D W, T W=W−C Velocity update −1 T D Π. 24c) The key to reduce complexity is to replace C by a matrix simpler to invert, which, however, is “similar” to C, in a sense that we will make precise. This technique is called inexact factorisation. 23) by an approximation A∗ obtained by replacing in the LU factorisation the matrix C −1 by convenient approximations, which we indicate by H1 and H2 , that is A∗ = L∗ U ∗ = C D 0 −DH1 D T = C D CH2 D T D(H2 − H1 )D T H2 D T IΠ IW D . 25) S ECTION 14 53 If we choose H2 = H1 , the discrete continuity equation is unaltered, that means that the approximated system still guarantees mass conservation at discrete level.
Some flow instabilities may occur only at the exit of the aortic valve and limited to the systolic phase. Indeed, in this region the Reynolds number may reach the value of few thousands only for the portion of the cardiac cycle corresponding the peak systolic velocity. Therefore, there is no sufficient time for a full turbulent flow to develop. , in the presence of a stenotic artery. The increase of the velocity at the location of the vessel restriction may induce turbulence to develop. This fact could explain the high increase in the noise caused by the blood stream in this situation.