Hessian‐based model reduction: large‐scale inversion and prediction

Hessian‐based model reduction was previously proposed as an approach in deriving reduced models for the solution of large‐scale linear inverse problems by targeting accuracy in observation outputs. A control‐theoretic view of Hessian‐based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large‐scale ( degrees of freedom) three‐dimensional contaminant transport problem in an urban environment, an application that requires real‐time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor speedup in computing time for the same level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

A stabilized finite element method for transient Navier–Stokes equations based on two local Gauss integrations

On the basis of two local Gauss integrations, a stabilized finite element method for transient Navier–Stokes equations is presented, which is defined by the lowest equal‐order conforming finite element subspace such as (or ) elements. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. The diffusion term in these equations is discretized by using finite element method, and the temporal differentiation and advection terms are treated by characteristic schemes. Moreover, we present some numerical simulations to demonstrate the effectiveness, good stability, and accuracy properties of our method. Especially, the rate of convergence study tells us that the stability still keeps well when the Reynolds number is increasing. Copyright © 2011 John Wiley & Sons, Ltd.     

Axisymmetric boundary integral formulation for a two‐fluid system

A 3D axisymmetric Galerkin boundary integral formulation for potential flow is employed to model two fluids of different densities, one fluid enclosed inside the other. The interface variables are the velocity potential and the normal velocity, and they can be solved for separately, the second linear system being symmetric. The algorithm is validated by comparing with the analytic solutions for a static interior spherical drop over a range of values for the relative densities of exterior and interior fluids and various boundary conditions. For time‐dependent simulations utilizing a level set method for the interface tracking, the accuracy has been checked by comparing against the known oscillation frequency of the sphere. Pinch‐off profiles corresponding to an initial two‐lobe geometry drop and D = 6 are also presented. Published in 2011 by John Wiley & Sons, Ltd.     

Stability and consistency of nonhydrostatic free‐surface models using the semi‐implicit θ‐method

The θ‐method is a popular semi‐implicit finite‐difference method for simulating free‐surface flows. Problem stiffness, arising because of the presence of both fast and slow timescale processes, is easily handled by the θ‐method. In most ocean, coastal, and estuary modeling applications, stiffness is caused by fast surface gravity wave timescales imposed on slower timescales of baroclinic variability. The method is well known to be unconditionally stable for shallow water (hydrostatic) models when , where θ is the implicitness parameter. In this paper, we demonstrate that the method is also unconditionally stable for nonhydrostatic models, when for both pressure projection and pressure correction methods. However, the methods result in artificial damping of the barotropic mode. In addition to investigating stability, we also estimate the form of artificial damping induced by both the free surface and nonhydrostatic pressure solution methods. Finally, this analysis may be used to estimate the damping or growth associated with a particular wavenumber and the overall order of accuracy of the discretization. Copyright © 2012 John Wiley & Sons, Ltd.     

Fast Ewald summation for Stokesian particle suspensions

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to , where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Fujiwhara interaction of tropical cyclone scale vortices using a weighted residual collocation method

The fundamental interaction between tropical cyclones was investigated through a series of water tank experiements by Fujiwhara  6 - 8 . However, a complete understanding of tropical cyclones remains an open research challenge although there have been numerous investigations through measurments with aircrafts/satellites, as well as with numerical simulations. This article presents a computational model for simulating the interaction between cyclones. The proposed numerical method is presented briefly, where the time integration is performed by projecting the discrete system onto a Krylov subspace. The method filters the large scale fluid dynamics using a multiresolution approximation, and the unresolved dynamics are modeled with a Smagorinsky type subgrid scale parameterization scheme. Numerical experiments with Fujiwhara interactions are considered to verify modeling accuracy. An excellent agreement between the present simulation and a reference simulation at has been demonstrated. At , the kinetic energy of cyclones is seen consolidated into larger scales with concurrent enstrophy cascade – suggesting a steady increase of energy containing scales – a phenomena that is typical in two‐dimensional turbulence theory. The primary results of this article suggest a novel avenue for addressing some of the computational challenges of mesoscale atmospheric circulations. Copyright © 2015 John Wiley & Sons, Ltd.     

A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition

A modified weakly compressible smoothed particle hydrodynamics (WCSPH) is presented, which utilizes consistent discretization schemes for spatial derivatives in the flow equations. Here, each SPH particle is considered as a computational point that represents a specific part of the fluid. To overcome non‐physical oscillations that usually arise in standard WCSPH, we modified the mass conservation equation by using a numerical filter. This modification is based on the difference between two discretization schemes used for the term . Furthermore, a new implementation of wall boundary condition in SPH is introduced. This condition is imposed on the pressure of wall boundary particles to ensure that the acceleration of each boundary particle in normal direction to the wall is zero. Thus, no penetration through walls will occur. To examine the performance of the modified method, we solved a series of two‐dimensional incompressible internal flow benchmark problems. By comparing the result with analytical solutions and the results of the standard WCSPH, we show that the use of consistent schemes in conjunction with the proposed numerical filter improves both accuracy and speed of the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time‐stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well‐known LBB‐stable finite element pair of third‐order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka‐like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.     

Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P₁ and P?P₁ mixed formulations lack convergence for inviscid flows. P?P₂ pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.     

Transition bifurcation branches in non-linear water waves

We are concerned with the numerical computation of progressive free surface gravity waves on a horizontal bed. They are regarded as families of bifurcation branches (λ,A)_Q of constant discharge Q. Numerically we determine two transition values Q₁ and Q₂ with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B₁, B₂ and B₃. Their members are families of waves (λ,A)_Q satisfying the conditions 0<Q² ?Q, Q <Q² ?Q and Q <Q² <B/27, respectively. The bifurcation patterns are analysed in some detail from the computed bifurcation diagram, which shows that in B₁ bifurcation is to the left and the amplitude A increases as the wavelength λ decreases; in B₂ bifurcation is to the right and turning points are observed nearly at breaking point. In B₃ bifurcation is to the right and A increases monotonically with λ.     

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of , where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two‐dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76× the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8× slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

A high‐order PIC method for advection‐dominated flow with application to shallow water waves

A hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.     

Multivariant finite elements for three-dimensional simulation of viscous incompressible flows

Within multivariant elements, which have restricted degrees of freedom at some nodes, different velocity components have different variations. Shape functions for the multivariant elements Q P_o and R P_o are developed. With such shape functions the value of a velocity component within a multivariant element is shown to depend upon all the independent components of velocity at the nodes of the element. The use of the Q₁ P₀ element to simulate flows with discontinuous boundary conditions generated disturbance throughout the flow domain, giving erroneous pressure and velocity distributions. The Q P_o element restricted the disturbance due to such discontinuities to a small region near the singular points, whereas the P P_o element completely eliminated the fluctuations. Flows with discontinuous boundary conditions were simulated with reasonable accuracy by partially relaxing the no-slip condition on the Q₁ P_o elements near the singular points.     

A dissipative particle dynamics study of flow in periodically grooved nanochannels

The effect of periodic rectangular wall roughness on planar nanochannel flow is investigated by dissipative particle dynamics simulation. The wall protrusion length is varied, and its effect on the flow is examined. Analysis of particle trajectories and average residence time reveals temporary trapping of fluid particles inside the rectangular cavities for a considerable amount of time. This trapping affects the density, velocity, pressure, and temperature distribution inside and close to the cavities. Inside the cavities, low‐velocity regions and regions of high density related to high pressure and high temperature are observed. When compared with that of the channel with flat walls case, lower flow velocities, temperatures, and pressures are observed for grooved channels. The reduction of the above quantities is more pronounced as the protrusion length, that is, the roughness characteristic length, decreases. Finally, the relation of friction factor, f, with the flow Reynolds number is discussed. The model predicts = constant in the range . The results of this work are of direct relevance to the design of nanofluidic devices. Copyright © 2011 John Wiley & Sons, Ltd.     

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

This paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least‐squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax–Milgram theorem, and besides proving the well‐posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart–Thomas elements of any order for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle‐bubble and edge‐bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators and showing the behavior of the associated adaptive algorithms, are reported. Copyright © 2015 John Wiley & Sons, Ltd.     

A computational methodology for two‐dimensional fluid flows

A weighted residual collocation methodology for simulating two‐dimensional shear‐driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two‐dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of a two‐dimensional flow in primitive variables. An excellent result has been observed—on resolving a shear layer and on the conservation of the potential and the kinetic energies—with respect to previously reported benchmark simulations. In particular, the shear‐driven simulation at CFL = 2.5 (Courant–Friedrichs–Lewy) and (Reynolds number) exhibits a linear speed up of CPU time with an increase of the time step, Δt. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.     

The behaviour of several stress–velocity–pressure mixed finite elements for Newtonian flows

The Q₂/P₁, P/P₁, P₂/P₀ and Q₁/P₀ velocity–pressure mixed elements are extended to the stress–velocity–pressure formulation, using the same interpolants for stress and velocity, and tested in the 4-to-1 contraction problem for Stokes flow. The comparison shows significant differences among them, which are not present when the velocity–pressure formulation is used. To provide a better understanding of the phenomenon, several variants of the previous elements are introduced, obtained by either changing the pressure space or by enriching the stress space with bubble functions. The formulation exhibits a strong sensitivity to the first alternative, while the second produces only a minor effect. These observations are confirmed by a convergence test effected on a regular problem with the explicit analytical solution. Also, as a result of the whole comparison, the P/P/P₁ element looks promising for three-field calculations.     

Three‐dimensional numerical simulation of red blood cell motion in Poiseuille flows

An immersed boundary method based on an FEM has been successfully combined with an elastic spring network model for simulating the dynamical behavior of a red blood cell (RBC) in Poiseuille flows. This elastic spring network preserves the biconcave shape of the RBC in the sense that after the removal of the body force for driving the Poiseuille flow, the RBC with its typical parachute shape in a tube does restore its biconcave resting shape. As a benchmark test, the relationship between the deformation index and the capillary number of the RBCs flowing through a narrow cylindrical tube has been validated. For the migration properties of a single cell in a slit Poiseuille flow, a slipper shape accompanied by a cell membrane tank‐treading motion is obtained for Re , and the cell mass center is away from the center line of the channel due to its asymmetric slipper shape. For the lower Re ?0.0137, an RBC with almost undeformed biconcave shape has a tumbling motion. A transition from tumbling to tank‐treading happens at the Reynolds number between 0.0137 and 0.03. In slit Poiseuille flow, the RBC can also exhibit a rolling motion like a wheel during the migration when the cell is released in the fluid flow with φ = π/2 and θ = π/2 (see Figure 12 for the definition of φ and θ). The lower the Reynolds number, the longer the rolling motion lasts; but the equilibrium shape and position are independent from the cell initial position in the channel. Copyright © 2014 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Reynolds‐stress modelling of M = 2.25 shock‐wave/turbulent boundary‐layer interaction

M = 2.25 shock‐wave/turbulent‐boundary‐layer interactions over a compression ramp for several angles (8, 13 and 18°) at Reynolds‐number Re=7 × 10³ were simulated with three low‐Reynolds second‐moment closures and a linear low‐Reynolds standard k–ε model. A detailed assessment of the turbulence closures by comparison with both mean‐flow and turbulent experimental quantities is presented. The Reynolds‐stress model which is wall‐topology free and which uses an optimized redistribution closure, is in good agreement with experimental data both for wall‐pressure and mean‐velocity profiles. Detailed analysis of three components of the Reynolds‐stress tensor (comparison with measurements and transport‐equation budgets) provides a critical evaluation of full Reynolds‐stress models for the separated supersonic compression ramp. The discrepancy observed in the shock‐wave foot region, between computations and measurements for the Reynolds‐stresses profiles, could be explained by considering the experimental shock‐wave oscillation and directions for future modelling work are indicated. Copyright © 2007 John Wiley & Sons, Ltd.