We present an LES-type variational multiscale theory of turbulence. Our approach derives completely from the incompressible Navier–Stokes equations and does not employ any
ad hoc devices, such as ...eddy viscosities. We tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows. In the calculations, we employed linear, quadratic and cubic NURBS. A dispersion analysis of simple model problems revealed NURBS elements to be superior to classical finite elements in approximating advective and diffusive processes, which play a significant role in turbulence computations. The numerical results are very good and confirm the viability of the theoretical framework.
The objective of this paper is to show that use of the element-vector-based definition of stabilization parameters, introduced in T.E. Tezduyar, Computation of moving boundaries and interfaces and ...stabilization parameters, Int. J. Numer. Methods Fluids 43 (2003) 555–575; T.E. Tezduyar, Y. Osawa, Finite element stabilization parameters computed from element matrices and vectors, Comput. Methods Appl. Mech. Engrg. 190 (2000) 411–430, circumvents the well-known instability associated with conventional stabilized formulations at small time steps. We describe formulations for linear advection–diffusion and incompressible Navier–Stokes equations and test them on three benchmark problems: advection of an L-shaped discontinuity, laminar flow in a square domain at low Reynolds number, and turbulent channel flow at friction-velocity Reynolds number of 395.
In this work, we combine (i) NURBS-based isogeometric analysis, (ii) residual-driven turbulence modeling and iii) weak imposition of no-slip and no-penetration Dirichlet boundary conditions on ...unstretched meshes to compute wall-bounded turbulent flows. While the first two ingredients were shown to be successful for turbulence computations at medium-to-high Reynolds number I. Akkerman, Y. Bazilevs, V. M. Calo, T. J. R. Hughes, S. Hulshoff, The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech. 41 (2008) 371–378; Y. Bazilevs, V.M. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, G. Scovazzi, Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Engrg., 197 (2007) 173–201, it is the weak imposition of no-slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher Reynolds number Y. Bazilevs, T.J.R. Hughes. Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput. Fluids 36 (2007) 12–26; Y. Bazilevs, C. Michler, V.M. Calo, T.J.R. Hughes, Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Engrg. 196 (2007) 4853–4862. These three ingredients form a basis of a possible practical strategy for computing engineering flows, somewhere between RANS and LES in complexity. We demonstrate this by solving two challenging incompressible turbulent benchmark problems: channel flow at friction-velocity Reynolds number 2003 and flow in a planar asymmetric diffuser. We observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data. This lends some credence to the proposed approach, which we believe may become a viable engineering tool.
We study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis. The problem of choice is the canonical Laplace equation in three dimensions. From this ...study we conclude that for a fixed number of unknowns and polynomial degree of approximation, a higher degree of continuity k drastically increases the CPU time and RAM needed to solve the problem when using a direct solver. This paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause.
We present a fully-coupled monolithic formulation of the fluid-structure interaction of an incompressible fluid on a moving domain with a nonlinear hyperelastic solid. The arbitrary ...Lagrangian–Eulerian description is utilized for the fluid subdomain and the Lagrangian description is utilized for the solid subdomain. Particular attention is paid to the derivation of various forms of the conservation equations; the conservation properties of the semi-discrete and fully discretized systems; a unified presentation of the generalized-
α
time integration method for fluid-structure interaction; and the derivation of the tangent matrix, including the calculation of shape derivatives. A NURBS-based isogeometric analysis methodology is used for the spatial discretization and three numerical examples are presented which demonstrate the good behavior of the methodology.
The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency ...range. This paper presents a new method with
no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the
optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with
L
2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial
hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.
In turbulence applications, strongly imposed no-slip conditions often lead to inaccurate mean flow quantities for coarse boundary-layer meshes. To circumvent this shortcoming, weakly imposed ...Dirichlet boundary conditions for fluid dynamics were recently introduced in Y. Bazilevs, T.J.R. Hughes, Weak imposition of Dirichlet boundary conditions in fluid mechanics, Comput. Fluids 36 (2007) 12–26. In the present work, we propose a modification of the original weak boundary condition formulation that consistently incorporates the well-known “law of the wall”. To compare the different methods, we conduct numerical experiments for turbulent channel flow at Reynolds number 395 and 950. In the limit of vanishing mesh size in the wall-normal direction, the weak boundary condition acts like a strong boundary condition. Accordingly, strong and weak boundary conditions give essentially identical results on meshes that are stretched to better capture boundary layers. However, on uniform meshes that are incapable of resolving boundary layers, weakly imposed boundary conditions deliver significantly more accurate mean flow quantities than their strong counterparts. Hence, weakly imposed boundary conditions present a robust technique for flows of industrial interest, where optimal mesh design is usually not feasible and resolving boundary layers is prohibitively expensive. Our numerical results show that the formulation that incorporates the law of the wall yields an improvement over the original method.
In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems.We stabilize the method via residual ...minimization. We discretize the problem using B-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor product mesh. We construct the solution over a smooth subspace of the residual. We can specify the solution subspace by reducing the polynomial order, by increasing the continuity, or by a combination of these. The Gramm matrix for the residual minimization method is approximated by a weighted H1 norm, which we can express as Kronecker products, due to the tensor-product structure of the approximations. We use the Gramm matrix as a preconditional which can be applied in a computational cost proportional to the number of degrees of freedom in 2D and 3D. Building on these approximations, we construct an iterative algorithm. We test the residual minimization method on several numerical examples, and we compare it to the Discontinuous Petrov–Galerkin (DPG) and the Streamline Upwind Petrov–Galerkin (SUPG) stabilization methods. The iGRM method delivers similar quality solutions as the DPG method, it uses smaller grids, it does not require breaking of the spaces, but it is limited to tensor-product meshes. The computational cost of the iGRM is higher than for SUPG, but it does not require the determination of problem specific parameters.
•We apply a stabilized mixed method based on residual minimization to isogeometric analysis.•We propose conjugate gradients iterative solver with alternating directions preconditioner.•We call the resulting method the isogeometric residual minimization (iGRM).•We test our method on several stationary advection–diffusion problems.•We compare against DPG and SUPG methods on model Eriksson–Johnson problem.
•The phase-field crystal equation is solved using a finite element discretization.•A mass-conserving, energy-stable, second-order time discretization is developed.•The results are proved rigorously, ...and verified numerically.•The implementation is done in PetIGA, an open source isogeometric analysis framework.•Three dimensional results showcase the robustness of the method.
The phase-field crystal equation, a parabolic, sixth-order and nonlinear partial differential equation, has generated considerable interest as a possible solution to problems arising in molecular dynamics. Nonetheless, solving this equation is not a trivial task, as energy dissipation and mass conservation need to be verified for the numerical solution to be valid. This work addresses these issues, and proposes a novel algorithm that guarantees mass conservation, unconditional energy stability and second-order accuracy in time. Numerical results validating our proofs are presented, and two and three dimensional simulations involving crystal growth are shown, highlighting the robustness of the method.
We use the alternating direction method to simulate implicit dynamics. Our spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we ...use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness matrices. As a result of this algebraic transformation, we can factorize the resulting system of equations in linear (i.e., O(N)) time at each step of the implicit method. We demonstrate the performance of our method in the model P-wave propagation problem. We then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods. We prove theoretically and experimentally the unconditional stability of both methods.
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