The time-dependent Navier-Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity-pressure-vorticity-temperature-heat-flux (u-P-w-T-q) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the 12-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to lo6, lid-driven cavity flow at Reynolds numbers up to lo4 and flow over a square obstacle at Reynolds number 200, are presented to validate the method.
The concepts and classification are brought forth for the strong-discontinuous interface, the weak-discontinuous interface, the micro-discontinuous interface and the all-continuous interface. The mechanical model is established for the dynamic fracture problem of the weak-discontinuous interface between a FGM coating and a FGM substrate. The Cauchy singular integral equation for the crack is derived by integral transform, and the allocation method is used to get the numerical solution. Analysis of the numerical solution indicates that the weak discontinuity is an important factor affecting the SIFs of the interfacial crack. To reduce the weak discontinuity is beneficial to the decrease of the SIFs. Contrast between the solution of the weak-discontinuous interface and that of the micro-discontinuous one shows that the micro-discontinuity is a kind of connection relation of mechanical property better than the weak discontinuity for the coating-substrate structure. To make the interface be micro-discontinuous is helpful to enhance the capacity of the functionally gradient coating-substrate interface to resist dynamic fracture. The first rank micro-discontinuity is enough to reduce the SIFs notably, however, the higher-rank micro-discontinuous terms, which is equal to or higher than the second rank, have less effect on the SIFs. In addition, the thickness of the coating and the substrate and the applied peel stress are also important factors affecting the dynamic SIFs.
SUMMARYA time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time-accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.
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