By Wolfgang Bangerth
Textual content compiled from the fabric provided through the second one writer in a lecture sequence on the division of arithmetic of the ETH Zurich in the course of the summer season time period 2002. thoughts of 'self-adaptivity' within the numerical resolution of differential equations are mentioned, with emphasis on Galerkin finite aspect versions. Softcover.
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Extra resources for Adaptive finite element methods for differential equations
The computation of the VAF then requires the specification of time dependence of the frictional coefficients. Recall that according to numerical simulations, in case of a simple dense fluid the behavior of K(t) should involve two relaxation times: one characteristic of the short time dynamics and one describing the long-time behavior. Quite a different type of model was introduced by Zwanzig and Bixon (1970). The physical picture is that of a single tagged particle moving through a viscoelastic continuum; its motion is described by the Langevin equation with the frequencydependent friction coefficient determined from hydrodynamic equations.
For example, when a revolving rod is immersed vertically into a liquid, the liquid surface near the rod is lifted up if the liquid is Newtonian and is drawn down if it is polymeric. A rotation of a disk at the bottom of a cylindrical vessel leads the Newtonian liquid to move out of the center, while the polymeric one moves towards the center. When flowing out of a vertical tube, the Newtonian liquid flow converges gradually at the diameter, while polymeric liquid quickly extends (more than two times) at first, and only after then it starts converging.
1 Hereditary behavior of a Newtonian liquid (a) versus Polymeric liquid (b). One of the main problems in polymeric dynamics is to find the appropriate expression for stress tensor, which would replace the Newtonian law for viscous liquid. Authors of the paper (Bird and Curtiss, 1984) proposed the next formula for polymeric melt of molecules in the form of freely connected chains consisting of N balls and N − 1 links of length a: 26 1 Heredity and Nonlocality 1 δi j − 3 pi j = pδi j + nNkT t −∞ μ (t − t )Ai j (t )dt − ε t −∞ ν (t − t )Bi j (t )dt , here Ai j and Bi j are tensors dependent on the finite deformation tensor and its velocity, n is polymeric molecules concentration, ν (t) = (16/π 2λ ) ∑ l −2 exp(−π l 2t/λ ), l and λ dν .
Adaptive finite element methods for differential equations by Wolfgang Bangerth