Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of hybrid Trefftz finite element method to non‐linear problems of minimal surface

This investigation provides a hybrid Trefftz finite element approach for analysing minimal surface problems. The approach is based on combining Trefftz finite element formulation with radial basis functions (RBF) and the analogue equation method (AEM). In this method, use of the analogue equation approach avoids the difficulty of treating the non‐linear terms appearing in the soap bubble equation, making it possible to solve non‐linear problems with the Trefftz method. Global RBF is used to approximate the inhomogeneous term induced from non‐linear functions and other loading terms. Finally, some numerical experiments are implemented to verify the efficiency of this method. Copyright © 2006 John Wiley & Sons, Ltd.     

Momentum Conservative Schemes for Shallow Water Flows

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.     

Parallel Finite Element Method and Time Stepping Control for Non-Isothermal Poro-Elastic Problems

This work focuses on parallel finite element simulation of thermal hydraulic and mechanical (THM) coupled processes in porous media, which is a common phenomenon in geological applications such as nuclear waste repository and CO2 storage facilities. The Galerkin finite element method is applied to solve the derived partial differential equations. To deal with the coupling terms among the equations, the momentum equation is solved individually in a monolithic manner, and moreover their solving processes are incorporated into the solving processes of nonisothermal hydraulic equation and heat transport equation in a staggered manner. The computation task arising from the present method is intensive if the method is applied to model a real geological application. Therefore, we present a parallel finite element method and a time stepping method with PI (proportional and integral feedback) automatic control to improve the computation efficiency. For parallel computing, the domain decomposition method is unitized to partition both computation tasks of the equation assembly and the linear solve, and the establishment of a global system of equations is thoroughly avoided. Moreover, an object-oriented concept of sparse matrix and iterative linear solver for large scale parallel and sequential simulation is developed. By simulating a real application with THM coupled processes, we show that the present parallel finite element method works fine for both monolithic and staggered scheme within coupling iterations, and furthermore we show the efficiency of the present method by the speedup we have achieved in the simulation.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical modeling of projectile penetration into compressible rigid viscoplastic media

We develop computational methods for modeling penetration of a rigid projectile into porous media. Compressible rigid viscoplastic models are used to capture the solid–fluid transition in behavior at high strain rates and account for damage/plasticity couplings and viscous effects that are observed in geological and cementitious materials. A hybrid time discretization is used to model the non‐stationary flow of the target material and the projectile–target interaction, i.e. an explicit Euler method for the projectile equation and a forward (implicit) method for the target boundary value problem. At each time step, a mixed finite element and finite‐volume strategy is used to solve the ‘target’ boundary value problem. Specifically, the non‐linear variational inequality for the velocity field is discretized using the finite element method while a finite‐volume method is used for the hyperbolic mass conservation and damage evolution equations. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented Lagrangian method is adopted. Numerical simulations of penetration into concrete were performed. By conducting a time step sensitivity study, it was shown that the numerical model is robust and computationally inexpensive. For the constants involved in the model (shear and volumetric viscosities, cut‐off yield limit, and exponential weakening parameter for friction) that cannot be determined from data, a parametric study was performed. It is shown that using the material model and numerical algorithms that developed the evolution of the density changes around the penetration tunnel, the shape and location of the rigid/plastic boundary, the compaction zones, and the extent of damage due to air‐void collapse are described accurately. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite difference solution of non‐linear systems of radiative–conductive heat transfer equations

A finite difference solution for a system of non‐linear integro–differential equations modelling the steady‐state combined radiative–conductive heat transfer is proposed. A new backward–forward finite difference scheme is formulated for the Radiative Transfer Equation. The non‐linear heat conduction equation is solved using the Kirchhoff transformation associated with a centred finite difference scheme. The coupled system of equations is solved using a fixed‐point method, which relates to the temperature field. An application on a real insulator composed of silica fibres is illustrated. The results show that the method is very efficient. Copyright © 2002 John Wiley & Sons, Ltd.     

A thermo‐hydro‐mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation

In this paper, the non‐isothermal elasto‐plastic behaviour of multiphase geomaterials in dynamics is investigated with a thermo‐hydro‐mechanical model of porous media. The supporting mathematical model is based on averaging procedures within the hybrid mixture theory. A computationally efficient reduced formulation of the macroscopic balance equations that neglects the relative acceleration of the fluids, and the convective terms is adopted. The modified effective stress state is limited by the Drucker–Prager yield surface. Small strains and dynamic loading conditions are assumed. The standard Galerkin procedure of the finite element method is applied to discretize the governing equations in space, while the generalized Newmark scheme is used for the time discretization. The final non‐linear set of equations is solved by the Newton method with a monolithic approach. Coupled dynamic analyses of strain localization in globally undrained samples of dense and medium dense sands are presented as examples. Vapour pressure below the saturation water pressure (cavitation) develops at localization in case of dense sands, as experimentally observed. A numerical study of the regularization properties of the finite element model is shown and discussed. A non‐isothermal case of incipient strain localization induced by temperature increase where evaporation takes place is also analysed. Copyright © 2016 John Wiley & Sons, Ltd.     

The coupled radial–axial deformations analysis of flexible pipes conveying fluid

This paper presents large deformation analysis of pipes conveying fluid in which two complicated behaviours are taken into consideration. The first is the coupling between radial and axial deformations of pipe wall, and the other is the interaction between a deformed pipe and transported fluid having the variable internal flow velocity. The coupled radial–axial deformation theory of the pipes and the continuity theory of flow inside the moving deformed pipes are developed to undertake these coupling behaviours. All strong and weak forms of governing equations are obtained by carrying out the virtual work formulation. The hybrid‐finite element method is used to solve the highly non‐linear static problems, which configure the initial large deflection and large strain conditions of the pipes. The state‐space finite element model for use in analyses of non‐linear vibration and system stability is established as well as the suggested numerical solution procedures. The numerical studies of the pipes under circumstances of intense radial loads such as deep‐water risers demonstrate that even a slight change of the radial deformation has a significant effect in increasing non‐linear responses, and reducing stabilities of the pipes. Copyright © 2004 John Wiley & Sons, Ltd.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Smooth finite strain plasticity with non‐local pressure support

The aim of this work is to introduce an alternative framework to solve problems of finite strain elastoplasticity including anisotropy and kinematic hardening coupled with any isotropic hyperelastic law. After deriving the constitutive equations and inequalities without any of the customary simplifications, we arrive at a new general elasto‐plastic system. We integrate the elasto‐plastic algebraico‐differential system and replace the loading–unloading condition by a Chen–Mangasarian smooth function to obtain a non‐linear system solved by a trust region method. Despite being non‐standard, this approach is advantageous, since quadratic convergence is always obtained by the non‐linear solver and very large steps can be used with negligible effect in the results. Discretized equilibrium is, in contrast with traditional approaches, smooth and well behaved. In addition, since no return mapping algorithm is used, there is no need to use a predictor. The work follows our previous studies of element technology and highly non‐linear visco‐elasticity. From a general framework, with exact linearization, systematic particularization is made to prototype constitutive models shown as examples. Our element with non‐local pressure support is used. Examples illustrating the generality of the method are presented with excellent results. Copyright © 2009 John Wiley & Sons, Ltd.     

Hybrid time domain solvers for the Maxwell equations in 2D

We present a new approach to time domain hybrid schemes for the Maxwell equations. By combining the classical FD‐TD scheme with two unstructured solvers, one explicit finite volume solver and one implicit finite element solver, we achieve a very efficient and flexible second‐order scheme. The second‐order accuracy of the hybrid scheme is verified through convergence studies on perfectly conducting as well as dielectric and diamagnetic circular cylinders. The numerical results also show its superiority to the FD‐TD scheme. Copyright © 2002 John Wiley & Sons, Ltd.     

The effect of grease on the activated sludge process in wastewater treatment

Abstract

The purpose of this paper is to introduce the major potentials and perspectives of applications of finite element analysis in solving the problems of shallow water wave equations. One‐dimensional and two‐dimensional shallow water wave equations will both be incorporated into the modeling procedures. For one‐dimensional flows, the models will cover the typical single channels, confluence channels system, division channels system, and natural river systems. As far as two‐dimensional flows are concerned, the overland flows are investigated. The simulation results are compared with the data obtained by physical modeling and field observation and with the results of other existing literature. The models were found to be very feasible in modeling the complex flow fields of shallow water wave equation problems.     

Dynamic inflation of non‐linear elastic and viscoelastic rubber‐like membranes

The present paper deals with the dynamic inflation of rubber‐like membranes.The material is assumed to obey the hyperelastic Mooney's model or the non‐linear viscoelastic Christensen's model. The governing equations of free inflation are solved by a total Lagrangian finite element method for the spatial discretization and an explicit finite‐difference algorithm for the time‐integration scheme. The numerical implementation of constitutive equations is highlighted and the special case of integral viscoelastic models is examined in detail. The external force consists in a gas flow rate, which is more realistic than a pressure time history. Then, an original method is used to calculate the pressure evolution inside the bubble depending on the deformation state. Our numerical procedure is illustrated through different examples and compared with both analytical and experimental results. These comparisons yield good agreement and show the ability of our approach to simulate both stable and unstable large strain inflations of rubber‐like membranes. Copyright © 2001 John Wiley & Sons, Ltd.     

Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization

We consider the numerical simulation of non‐linear multi‐body contact problems in elasticity on complex three‐dimensional geometries. In the case of warped contact boundaries and non‐matching finite element meshes, particular emphasis has to be put on the discretization of the transmission of forces and the non‐penetration conditions at the contact interface. We enforce the discrete contact constraints by means of a non‐conforming domain decomposition method, which allows for optimal error estimates. Here, we develop an efficient method to assemble the discrete coupling operator by computing the triangulated intersection of opposite element faces in a locally adjusted projection plane but carrying out the required quadrature on the faces directly. Our new element‐based algorithm does not use any boundary parameterizations and is also suitable for isoparametric elements. The emerging non‐linear system is solved by a monotone multigrid method of optimal complexity. Several numerical examples in 3D illustrate the effectiveness of our approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero‐velocities in solid material. The resulting parameter optimization problem is solved by a non‐linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design‐dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non‐physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady‐state counterparts. Copyright © 2011 John Wiley & Sons, Ltd.