A symmetric weak form of Biot's equations based on redundant variables representing the fluid,using a Helmholtz decomposition of the fluid displacement vector field

A novel symmetric weak formulation of Biot's equations for linear acoustic wave propagation in layered poroelastic media is presented. The primary variables used are the frame displacement, the acoustic pore pressure, the scalar potential and the vector potential obtained from a Helmholtz decomposition of the fluid displacement. Also a symmetric weak form based on the frame displacement, the pore pressure and the fluid displacement is obtained as an intermediate result. hp finite element simulations of a double leaf partition based on this new weak formulation is verified against simulation results from the classical frame displacement, fluid displacement formulation and a frame displacement pore pressure formulation. All three formulations simulated, displays the same rate of convergence with respect to finite element bases polynomial degree. The novel formulation also extends a previously published frame displacement, pore pressure, scalar fluid displacement potential formulation with an implicit irrotational fluid displacement assumption to a full representation of Biot's equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Fundamental solution for a layered porous half space subject to a vertical point force or a point fluid source

The focus of this contribution is to develop a transmission and reflection matrices (TRM) method for a layered porous half-space subject to a point force or a fluid point source. Applying the Hankel and the Fourier transformation, the general solutions for the displacements, stresses and pore pressure are derived from the potentials for the solid skeleton and the pore fluid as well as the governing equations of Biots theory. The transmission and reflection matrices (TRM) for each interface are obtained by using the general solutions as well as the continuity conditions at the interface. The TRM method for the layered porous medium is developed on the basis of the transmission and reflection matrices (TRM) and the boundary conditions as well as the source terms for the point force or the fluid point source. The fundamental solutions of the point force and the point fluid source in both the frequency domain and the time domain are obtained by using the proposed TRM method. Some numerical examples are given in the paper.     

The continuous adjoint approach to the k–ε turbulence model for shape optimization and optimal active control of turbulent flows

The continuous adjoint to the incompressible Reynolds-averaged Navier–Stokes equations coupled with the low Reynolds number Launder–Sharma k–ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder–Sharma k–ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the ‘frozen turbulence assumption’, the gain in the optimization turnaround time offered by the proposed method is quantified.     

A SPACE–TIME FINITE ELEMENT METHOD FOR THE EXTERIOR STRUCTURAL ACOUSTICS PROBLEM: TIME-DEPENDENT RADIATION BOUNDARY CONDITIONS IN TWO SPACE DIMENSIONS

A time-discontinuous Galerkin space–time finite element method is formulated for the exterior structural acoustics problem in two space dimensions. The problem is posed over a bounded computational domain with local time-dependent radiation (absorbing) boundary conditions applied to the fluid truncation boundary. Absorbing boundary conditions are incorporated as ‘natural’ boundary conditions in the space–time variational equation, i.e. they are enforced weakly in both space and time. Following Bayliss and Turkel, time-dependent radiation boundary conditions for the two-dimensional wave equation are developed from an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a non-dimensional wavenumber. In this paper, we undertake a brief development of the time-dependent radiation boundary conditions, establishing their relationship to the exact impedance (Dirichlet-to-Neumann map) for the acoustic fluid, and characterize their accuracy when implemented in our space–time finite element formulation for transient structural acoustics. Stability estimates are reported together with an analysis of the positive form of the matrix problem emanating from the space–time variational equations for the coupled fluid-structure system. Several numerical simulations of transient radiation and scattering in two space dimensions are presented to demonstrate the effectiveness of the space–time method.     

Slip at a uniformly porous boundary: effect on fluid flow and mass transfer

An approximate solution to the 2-D Navier-Stokes equations for steady, isothermal, incompressible, laminar flow in a channel bounded by one porous wall subject to uniform suction is derived. The solution is valid for small values of the Reynolds number based on the suction velocity and channel height. Solute transport is considered numerically by decoupling the equations representing momentum and mass transfer. The effect of fluid slip at the porous boundary on the axial and transverse components of fluid velocity, axial pressure drop and mass transfer is investigated.     

A new boundary finite element method for fluid–structure interaction problems

In order to study problems on fluid–structure interaction, we have used a mixed formulation which couples the classical functional of the structure with a new variational formulation by integral equations for the fluid. This formulation has the advantage over the finite element methods of avoiding the discretization of the fluid domain. Furthermore, unlike collocation methods, the explicit calculation of the Hadamard finite part of the singular integrals is avoided. This leads after discretization by boundary finite elements to a small and symmetrical algebraic system. Typical examples are presented that demonstrate the efficiency of this variational formulation by studying the sound transmission through a baffled plane structure and through a flexible panel backed by a rigid cavity. These include the calculation of the transmission loss factor and the determination of which modes dominate the noise transmission. Good agreement is obtained between numerical results and analytical results found in the literature.     

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.     

Gravity wave interaction with porous structures in two-layer fluid

Oblique wave interaction with rectangular porous structures of various configurations in two-layer fluid are analyzed in finite water depth. Wave characteristics within the porous structure are analyzed based on plane wave approximation. Oblique wave scattering by a porous structure of finite width and wave trapping by a porous structure near a wall are studied under small amplitude wave theory. The effectiveness of three types of porous structures—a semi-infinite porous structure, a finite porous structure backed by a rigid wall, and a porous structure with perforated front and rigid back walls—in reflecting and dissipating wave energy are analyzed. The reflection and transmission coefficients for waves in surface and internal modes and the hydrodynamic forces on porous structures of the aforementioned configurations are computed for various physical parameters in two-layer fluid. The eigenfunction expansion method is used to deal with waves past the porous structure in two-layer fluid assuming the associated eigenvalues are distinct. An alternate procedure based on the Green’s function technique is highlighted to deal with cases where the roots of the dispersion relation in the porous medium coalesce. Long wave equations are derived and the dispersion relation is compared with that derived based on small amplitude wave theory. The present study will be of significant importance in the design of various types of coastal structures used in the marine environment for the reflection and dissipation of wave energy.     

An effective stress based numerical model for hydro-mechanical analysis in unsaturated porous media

 A fully coupled flow-deformation model is presented for the behaviour of unsaturated porous media. The governing equations are derived based on the equations of equilibrium, effective stress concept, Darcy's law, Henry's law, and the conservation of fluid mass. Macroscopic coupling between the flow and deformation fields is established through the effective stress parameters. The microscopic link between the volumetric deformations of the two pore system (i.e. the pore-air and the pore-water) is established using Betti's reciprocal theorem. Both links are essential for a proper modelling of flow and deformation in unsaturated porous media. The discretised form of the governing equations is obtained using the finite element technique. As application of the model, experimental results from several laboratory tests reported in the literature are modelled numerically. Good agreement is obtained between the numerical and the experimental results in all cases.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of \(u{-}p\) formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.     

Topology optimization of creeping fluid flows using a Darcy–Stokes finite element

A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no‐slip condition along the solid–fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy–Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no‐slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid–fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy–Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0–1 (void–solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element modelling of fibre reinforced polymer sandwich panels exposed to heat

A finite element model that predicts temperature distribution in a composite panel exposed to a heat source, such as fire, is described. The panel is assumed to be composed of skins consisting of polymer matrix reinforced with fibres and a lightweight core (the paper concentrates on the crucial aspect of the problem, i.e. the behaviour of the ‘hot’ skin of the panel. The core is assumed not to decompose, and the ‘cold’ skin is treated exactly as the ‘hot’ skin.) It is assumed that the polymer matrix undergoes chemical decomposition. Such a model results in a set of coupled non‐linear transient partial differential equations. A Galerkin finite element framework is formulated to yield a fully implicit time stepping scheme. The crucial input parameters for the model are carefully identified for subsequent experimental determination. Copyright © 2004 John Wiley & Sons, Ltd.     

A phase field model for the freezing saturated porous medium

In terms of the mixture theory and phase field theory, a phase field model is developed for the saturated porous medium undergoing phase transition. In the proposed model, it is postulated that during the phase transition of the porous medium, both the solid skeleton and pore fluid will undergo phase transition. The phase states of the solid skeleton and pore fluid are characterized by respective order parameters. To simplify the proposed phase field model, the temperatures and order parameters of the solid skeleton and pore fluid are assumed to be equal. The balance laws of the porous medium are given by the conventional mixture theory. The order parameter and the porosity of the porous medium are considered as internal variables and their evolution equations are determined by the entropy inequality of the porous medium. The constitutive representations for the stresses, entropies, heat fluxes, drag force and the evolution equations for the order parameter and porosity are derived by exploitation of the entropy inequality. To illustrate the proposed model, a concrete phase field model for the freezing porous medium is established in the paper. In the model, the memory effect associated with phase transition of the porous medium is taken into account by assuming Stieltjes integral for the strain energy of the porous medium. The constitutive representations for the above variables are then derived according to the proposed free energy expression for the porous medium. Finally, the boundary condition associated with the proposed model and the determination of some parameters involved in our model are discussed in the paper briefly.     

A two‐scale approach for fluid flow in fractured porous media

A two‐scale numerical model is developed for fluid flow in fractured, deforming porous media. At the microscale the flow in the cavity of a fracture is modelled as a viscous fluid. From the micromechanics of the flow in the cavity, coupling equations are derived for the momentum and the mass couplings to the equations for a fluid‐saturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two‐scale approach and integrated over time. By exploiting the partition‐of‐unity property of the finite element shape functions, the position and direction of the fractures is independent from the underlying discretization. The resulting discrete equations are non‐linear due to the non‐linearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach, and show that faults in a deforming porous medium can have a significant effect on the local as well as on the overall flow and deformation patterns. Copyright © 2006 John Wiley & Sons, Ltd.     

Weak,anisotropic symmetric formulations of Biot's equations for vibro‐acoustic modelling of porous elastic materials

In this paper a fully anisotropic symmetric weak formulation of Biot's equations for vibro‐acoustic modelling of porous elastic materials in the frequency domain is proposed. Starting from Biot's equations in their anisotropic form, a mixed displacement–pressure formulation is discussed in terms of Cartesian tensors. The anisotropic equation parameters appearing in the differential equations are derived from material parameters which are possible to determine through experimental testing or micro‐structural simulations of the fluid and the porous skeleton. Solutions are obtained by applying the finite element method to the proposed weak form and the results are verified against a weak displacement‐based formulation for a foam and plate combination. Copyright © 2010 John Wiley & Sons, Ltd.     

饱和两相介质近场波动问题的一种时域全显式数值计算方法

基于 u -p形式的饱和两相介质弹性波动方程,开展了饱和两相介质近场波动问题时域显式数值计算方法的研究。通过对波动方程中的质量矩阵和孔隙流体压缩矩阵进行对角化处理,消除了方程中的动力耦联,实现了波动方程的解耦。分别应用中心差分法和Newmark常平均加速度法求解固相位移和速度,基于向后差分法求解孔隙流体压力,推导得到了饱和两相介质动力响应的时域显式逐步积分的计算列式,建立了饱和两相介质近场波动问题的一种新的时域全显式数值计算方法。进行了该文方法中矩阵对角化合理性的验证。将该方法的数值解与相应的解析解进行对比,二者符合良好,验证了该方法的正确性。将该文建立的时域数值计算方法与透射人工边界方法相结合,应用于饱和两相介质的近场波动问题,进行了饱和土场地地震响应的计算研究,计算结果符合弹性波动理论的基本规律,表明该方法对于饱和两相介质近场波动问题时域计算求解的适用性。基于该方法中时域递推计算格式的传递矩阵,进行了该方法稳定性特性的研究。该文建立的数值计算方法具有时域全显式算法的基本特征。方法中对动力响应的全部分量均采用递推和迭代的模式进行求解,避免了求解耦联的动力方程组。该方法具有较高的计算效率,是进行饱和两相介质近场波动问题时域计算求解的一种有效的算法。     

Flow of compressible fluid in porous elastic media

A finite element formulation for flow of fluid in a porous elastic media has been derived from a Gurtin type variational principle. Biot's field equations for porous media have been used in which the constitutive relations include the compressibility of the fluid. It has been shown that by proper choice of the form of the compressibility of the fluid as a function of the state variables, this option of the formulation can be used to treat the partially saturated soil. The method is general with respect to geometry, boundary conditions and material properties. Finally, the results of a series of examples have been presented and compared with exact results to demonstrate accuracy and applicability.     

A stable Galerkin reduced order model for coupled fluid–structure interaction problems

A stable reduced order model (ROM) of a linear fluid–structure interaction (FSI) problem involving linearized compressible inviscid flow over a flat linear von Kármán plate is developed. Separate stable ROMs for each of the fluid and the structure equations are derived. Both ROMs are built using the ‘continuous’ Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The mode shapes for the structure ROM are the eigenmodes of the governing (linear) plate equation. The fluid ROM basis is constructed via the proper orthogonal decomposition. For the linearized compressible Euler fluid equations, a symmetry transformation is required to obtain a stable formulation of the Galerkin projection step in the model reduction procedure. Stability of the Galerkin projection of the structure model in the standard L² inner product is shown. The fluid and structure ROMs are coupled through solid wall boundary conditions at the interface (plate) boundary. An a priori energy linear stability analysis of the coupled fluid/structure system is performed. It is shown that, under some physical assumptions about the flow field, the FSI ROM is linearly stable a priori if a stabilization term is added to the fluid pressure loading on the plate. The stability of the coupled ROM is studied in the context of a test problem of inviscid, supersonic flow past a thin, square, elastic rectangular panel that will undergo flutter once the non‐dimensional pressure parameter exceeds a certain threshold. This a posteriori stability analysis reveals that the FSI ROM can be numerically stable even without the addition of the aforementioned stabilization term. Moreover, the ROM constructed for this problem properly predicts the maintenance of stability below the flutter boundary and gives a reasonable prediction for the instability growth rate above the flutter boundary. Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling of porous piezoelectric structures by the meshless local Petrov-Galerkin method

A meshless method based on the local Petrov-Galerkin approach is proposed to solve initial-boundary value problems of porous piezoelectric solids. Constitutive equations for porous piezoelectric materials possess a coupling between mechanical displacements and electric intensity vectors in both solid and fluid phases. Stationary and transient 2-D and 3-D axisymmetric problems are considered in this article. Nodal points are spread on the problem domain, and each node is surrounded by a small circle for simplicity. The spatial variation of displacements and electric potentials for both phases is approximated by the moving least-squares scheme. After performing the spatial integration, one obtains a system of ordinary differential equations for certain nodal unknowns. The resulting system is solved numerically by the Houbolt finite-difference scheme as a time stepping method. The proposed method is applied to bending problems associated with a porous piezoelectric 2-D plate and 3-D axisymmetric cylinder under simply supported and clamped boundary conditions.