A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.     

Mixed finite element method for coupled thermo‐hydro‐mechanical process in poro‐elasto‐plastic media at large strains

A mixed finite element for coupled thermo‐hydro‐mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three‐filed variational principle. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto‐plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 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.     

Flow‐induced vibrations of non‐linear cables. Part 1: Models and algorithms

In this paper, we develop governing equations for non‐linear cables as well as a formulation for the coupled flow‐structure problem. The structure is discretized with second‐order accuracy while the flow is discretized using spectral/hp elements in the context of the arbitrary Lagrangian–Eulerian formulation (ALE). Several benchmark problems are considered and the computational implementation is detailed. In the second part of this work large‐scale simulation examples are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

高温下混凝土中热-湿-气-力学耦合过程数值模拟

对高温下混凝土中热-湿-气-力学耦合过程分析提出了一个多孔多相介质的非混溶-混溶两级数学模型。数学模型基于控制干空气、湿份及基质溶解物的质量守恒、混凝土介质混合体的动量守恒和焓(能量)守恒的耦合偏微分方程组。给出了模型的控制方程、状态方程与所采用的本构定律。发展了相应的非线性耦合问题的有限元数值分析过程,以数值模拟火灾和热辐射等热荷载作用下的热-湿-气-力学耦合行为,并进而分析所发生的破坏现象。数值结果例题显示所发展的数学模型和数值方法在重现高温下混凝土中热-湿-气-力学耦合行为的有效性。     

Finite element analysis of flows in secondary settling tanks

The equations governing unsteady flows in secondary settling tanks, a component of the wastewater treatment process, are analysed using the finite element method. The model corresponding to such liquid–solid flows is highly non‐linear and coupled, and incorporates the effects of turbulence. The results of numerical simulations are compared against experimental results from tests on full‐scale settling tanks, and against results obtained from a finite difference code based on an idealized one‐dimensional flux theory. The results compare well with the test results, over the range of applicability of the model. Copyright © 2005 John Wiley & Sons, Ltd.     

Computational simulation of chemical dissolution‐front instability in fluid‐saturated porous media under non‐isothermal conditions

This paper primarily deals with the computational aspects of chemical dissolution‐front instability problems in two‐dimensional fluid‐saturated porous media under non‐isothermal conditions. After the dimensionless governing partial differential equations of the non‐isothermal chemical dissolution‐front instability problem are briefly described, the formulation of a computational procedure, which contains a combination of using the finite difference and finite element method, is derived for simulating the morphological evolution of chemical dissolution fronts in the non‐isothermal chemical dissolution system within two‐dimensional fluid‐saturated porous media. To ensure the correctness and accuracy of the numerical solutions, the proposed computational procedure is verified through comparing the numerical solutions with the analytical solutions for a benchmark problem. As an application example, the verified computational procedure is then used to simulate the morphological evolution of chemical dissolution fronts in the supercritical non‐isothermal chemical dissolution system. The related numerical results have demonstrated the following: (1) the proposed computational procedure can produce accurate numerical solutions for the planar chemical dissolution‐front propagation problem in the non‐isothermal chemical dissolution system consisting of a fluid‐saturated porous medium; (2) the Zhao number has a significant effect not only on the dimensionless propagation speed of the chemical dissolution front but also on the distribution patterns of the dimensionless temperature, dimensionless pore‐fluid pressure, and dimensionless chemical‐species concentration in a non‐isothermal chemical dissolution system; (3) once the finger penetrates the whole computational domain, the dimensionless pore‐fluid pressure decreases drastically in the non‐isothermal chemical dissolution system.     

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 fully coupled finite element analysis of water‐table fluctuation and land deformation in partially saturated soils due to surface loading

A fully coupled numerical model is presented for the water‐table fluctuation and land deformation in partially saturated soils due to surface loading. This numerical model is developed based on the poroelastic governing equations for groundwater flow in deforming variably saturated porous media and the Galerkin finite element method. The numerical model is verified and validated against a one‐dimensional consolidation problem concerning surface loading on a soil column which has six different initial water‐table elevations. The numerical model is then applied to a two‐dimensional consolidation problem of surface loading on a partially saturated soil at a construction site. Results from the numerical simulations of both problems show that the water table fluctuates in the partially saturated soils, and the unsaturated zone above the water table has significant effects on the consolidation behaviour of the partially saturated soils under surface loading. Such effects are caused by the permanent absorption of a portion of the mechanical loading stress and the weak hydromechanical coupling between the solid skeleton deformation field and the groundwater flow field in the unsaturated zone due to its partial saturation. Copyright © 2000 John Wiley & Sons, Ltd.     

Modelling of reinforced polymer composites subject to thermo‐mechanical loading

A coupled finite element model is developed to analyse the thermo‐mechanical behaviour of a widely used polymer composite panel subject to high temperatures at service conditions. Thermo‐chemical and thermo‐mechanical models of previous researchers have been extended to study the thermo‐chemical decomposition, internal heat and mass transfer, deformation and the stress state of the material. The phenomena of heat and mass transfer and thermo‐mechanical deformation are simulated using three sets of governing equations, i.e. energy, gas mass diffusion and deformation equations. These equations are then assembled into a coupled matrix equation using the Bubnov–Galerkin finite element formulation and then solved simultaneously at each time interval. An experimentally tested 1.09 cm thick glass‐fibre woven‐roving/polyester resin composite panel is analysed using the numerical model. Results are presented in the form of temperature, pore pressure, deformation, strain and stress profiles and discussed. The maximum normal stress failure criterion is used in order to establish the load‐bearing capability of the composite panel. Significant pore gas pressure build‐ups (to 0.8 MPa and higher) have been perceived at high thermo‐chemical decomposition rates where the material experiences a complex expansion/contraction phenomenon. It is found that the composite panel experiences structural instability at elevated temperatures up to 300°C but retains its integrity even under moderate external loading. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.     

Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Lagrangian frame of reference using ρ, u,p primitive variables

In this paper, we present computations of the non‐weak solutions of class C¹¹ of transient Navier–Stokes equations for compressible flow in Lagrangian frame of reference using space‐time least squares finite element formulation with primitive variables ρ, u, p. For high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Compression of air in a rigid cylinder by a rigid, massless and frictionless piston is used as a model problem. True time evolutions of class C¹¹ are reported beginning with the first time step until steady shock conditions are achieved. Comparisons with analytical solutions are presented when possible. Copyright © 2001 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.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

Many finite elements exhibit the so‐called ‘volumetric locking’ in the analysis of incompressible or quasi‐incompressible problems.In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective–diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi‐implicit and explicit 2D and 3D non‐linear transient dynamic analysis of an impact problem and a bulk forming process are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of transient flow in pipelines with fluid–structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non‐linear, first‐order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 John Wiley & Sons, Ltd.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.