Dynamic analysis of a water–soil–pore water coupling system

This paper explores the possible integration of the dynamic analyses of interaction of pore fluid flow and porous structure. The generalized fluid–structure interaction (FSI) algorithm in ADINA is applied for seismic analysis of a reservoir–earth dam–foundation system. In the given numerical example, water in the reservoir interacts with the earth dam on the upstream slope of the dam, and the foundation at the reservoir bottom. Pore water couples with soil particles throughout the earth dam and foundation. The reservoir water is modeled using both the Navier–Stokes equation based fluid element and the subsonic potential based fluid element. In the coupled analysis, hydrodynamic pressures and velocities are presented in the reservoir zone, while displacements, pore pressures and stresses are given for the earth dam and foundation domains at any earthquake time. The generalized FSI model is of great significance in performing soil liquefaction analysis if an appropriate soil plasticity model is employed in accounting for cyclic behavior of soils.     

A finite element model for poroelastic beams with axial diffusion

The finite element method is used for those fluid-saturated poroelastic rods in which diffusion is possible only in the axial direction as a result of the microgeometry of the solid skeleton material. Variational principles are developed first for this purpose. Two types of variables, the displacements and pore pressure, are involved in the time dependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy’s law) into the Euler–Lagrange equations of the functionals. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solutions for both types of variables are found to coincide well with the existing analytical solutions. Some interesting results are demonstrated which are not available by analytical methods.     

A Fully Implicit Computational Strategy for Strongly Coupled Fluid–Solid Interaction

This article summarises the authors’ research work in the area of computational modelling of interaction of fluid flow with solid structures. Our approach relies on a fully implicit iterative solution strategy which resolves the strong coupling and allows for optimal rate of convergence of the residuals. Therefore, the methodology is a viable competitor for the solution of the highly nonlinear interaction of fluid flow with solid structures that experience large displacements and deformations. The key ingredients of our strategy include the following: Stabilised low order velocity–pressure finite elements are used for the modelling of the fluid flow combined with an arbitrary Lagrangian–Eulerian (ALE) strategy. For the temporal discretisation of both fluid and solid bodies, the discrete implicit generalised-α method is employed. An important aspect of the present work is the introduction of the independent interface discretisation, which allows an efficient, modular and expandable implementation of the solution strategy. A simple data transfer strategy based on a finite element type interpolation of the interface degrees of freedom guarantees kinematic consistency and equilibrium of the stresses along the interface. The resulting strongly coupled set of nonlinear equations is solved by means of a partitioned solution procedure, which is based on the Newton–Raphson methodology and incorporates the full linearisation of the overall incremental problem. Thus, asymptotically quadratic convergence of the residuals is achieved. Numerical examples are presented to demonstrate the robustness and efficiency of the methodology. Finally, we present the results obtained by combining the presented methodology with a remeshing procedure.     

Finite element MAC scheme in general curvilinear coordinates

A combined study of F.D. and F.E. methods for 2-D incompressible Navier-Stokes flows is undertaken. In primitive variable formulation, major difficulties are connected to spurious numerical oscillations which may arise from the enforcement of the incompressibility constraint. With regard to this problem, various F.D. schemes differ essentially according to the variable location on the mesh points, while F.E. schemes are analogously differentiated by the interpolation functions adopted for the different variables. In the present paper, we propose an F.E. analog of MAC scheme, which can be accomplished by different interpolation functions for the two velocity components. This new F.E. scheme-although based on low order approximations-eliminates all spurious oscillations. The extension to curvilinear quadrilateral elements—which is needed in order to achieve geometrical versatility—requires the problem to be formulated in general curvilinear coordinates and the contravariant velocity components to be assumed as variables. Some numerical results are presented and discussed, in order to assess the capabilities of the proposed model.     

Fluid-structure interaction study of the start-up of a rocket engine nozzle

The aim of this paper is to analyze the aeroelastic processes developed during the starting phase of a rocket engine via a coupling fluid/structure code. This analysis gives a better understanding of the behavior of the structure as the shock waves propagate inside the engine nozzle. The gasdynamics Euler equations are solved for the fluid and constitutive linear elastic solid assuming large displacements and rotations with no material damping is adopted for the structure. The coupling of each subproblem is carried out with a Gauß-Seidel algorithm over the fluid and structure states. For the fluid problem an ALE (Arbitrary Lagrangian-Eulerian) formulation is used. It allows us to define a reference system following the moving boundaries while the structure is deformed. The code is validated with a study of the flutter phenomena that may occur when a supersonic compressible fluid flows over a flat solid plate. Regarding the rocket engine ignition problem, a modal analysis of the structure is performed in order to analyze the eigenfrequency shifts when considering the coupling with the fluid flow.     

The influence of limit cycle topology on the phase resetting curve

Understanding the phenomenology of phase resetting is an essential step toward developing a formalism for the analysis of circuits composed of bursting neurons that receive multiple, and sometimes overlapping, inputs. If we are to use phase-resetting methods to analyze these circuits, we can either generate phase-resetting curves (PRCs) for all possible inputs and combinations of inputs, or we can develop an understanding of how to construct PRCs for arbitrary perturbations of a given neuron. The latter strategy is the goal of this study. We present a geometrical derivation of phase resetting of neural limit cycle oscillators in response to short current pulses. A geometrical phase is defined as the distance traveled along the limit cycle in the appropriate phase space. The perturbations in current are treated as displacements in the direction corresponding to membrane voltage. We show that for type I oscillators, the direction of a perturbation in current is nearly tangent to the limit cycle; hence, the projection of the displacement in voltage onto the limit cycle is sufficient to give the geometrical phase resetting. In order to obtain the phase resetting in terms of elapsed time or temporal phase, a mapping between geometrical and temporal phase is obtained empirically and used to make the conversion. This mapping is shown to be an invariant of the dynamics. Perturbations in current applied to type II oscillators produce significant normal displacements from the limit cycle, so the difference in angular velocity at displaced points compared to the angular velocity on the limit cycle must be taken into account. Empirical attempts to correct for differences in angular velocity (amplitude versus phase effects in terms of a circular coordinate system) during relaxation back to the limit cycle achieved some success in the construction of phase-resetting curves for type II model oscillators. The ultimate goal of this work is the extension of these techniques to biological circuits comprising type II neural oscillators, which appear frequently in identified central pattern-generating circuits.     

Utilisation de la méthode des éléments finis en mécanique des fluides,I

An analysis is made of the possibility of building divergence free elements for the approximation of the Navier-Stokes equations of an incomprensible fluid. Two types of appoximation are considered. In the first real divergence free elements are considered. Many examples are given and error bounds are derived. It is shown that 0 (h ⁴) precision can be obtained using fourth degree polynomials. In the second type of approximation the incompressibility condition is satisfied only in the average. Two and three dimensional examples are given, using respectively second and third order polynomials. A possible numerical scheme is proposed for the case the steady-state linearized Stokes problem. Experimental evidence shows that it can be also used in the general non-linear case. Simple numerical results are presented for the cavity problem in two dimensions.

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Ce travail a été rendu possible en partie grace à une subvention du Conseil National de la Recherche du Canada.     

弱可压缩流体边界处理算法

针对流体与固体边界的交互模拟问题,提出一种基于弱可压缩光滑粒子流体动力学(SPH)的边界处理算法.首先,引入一种新的体积权重函数,解决固体边界非均匀采样区域流体密度的计算误差问题;然后,提出一种新的边界力计算模型,避免校正流体粒子位置信息,保证固体边界不可穿透;最后,提出一种改进的流体压力计算模型,保证流体的弱可压缩性.实验结果表明,所提算法可以有效地解决基于位置校正的边界处理方法在模拟弱可压缩流体与非均匀采样固体边界交互时存在的稳定性问题,且仅需边界粒子的位置信息,在节约内存的同时避免了位置校正所带来的额外计算开销.     

Topology redesign for performance by large admissible perturbations

A methodology is developed for optimal structural topology design subject to several performance constraints. Eight-node solid elements are used to model the initial structure, which is a uniform solid block satisfying the boundary conditions and subjected to external loading. The Young modulus of each solid element or group of elements is used as redesign variable. A minimum change function is used as an optimality criterion. Performance constraints include static displacements, natural frequencies, forced response amplitudes, and static stresses. These constraints are treated by the large admissible perturbation methodology which makes it possible to achieve the performance objectives incrementally without trial and error or repetitive finite element analyses for changes in the order of 100–300%. Thus, the optimal topology is reached in about four to five iterations, where each iteration includes one finite element analysis and setting of an upper limit for the value of the modulus of elasticity to produce a manufacturable structure. Several numerical applications are presented using three different benchmark structures to demonstrate the methodology and the impact of performance constraints on the generated topology.     

An interpolation strategy for discrete-time bilinear MPC problems

Input-output (I-O) feedback linearization suffers from a number of restrictions which have limited its use in model-based predictive control. Some of these restrictions do not apply to the case of bilinear systems, but problems with input constraints and unstable zero dynamics persist. This paper overcomes these difficulties by means of an interpolation strategy. Involved in this interpolation is a feasible and stabilizing trajectory, which is computed through the use of invariant feasible sets, and a more aggressive trajectory, which can be chosen to be either the unconstrained optimal trajectory or any alternative one     

1D wave propagation in saturated viscous geomaterials: Improvement and validation of a fractional step Taylor–Galerkin finite element algorithm

This paper tackles the numerical simulation of 1D wave propagation in saturated viscous porous media, and especially in soil-like geomaterials. For this purpose, an improved fractional step Taylor–Galerkin algorithm is first formulated and then validated on the basis of a new analytical solution.The algorithm, based on a stress–velocity–pressure formulation of the hydro-mechanical problem, combines an explicit Taylor–Galerkin method with a fractional time-stepping, while an accurate Runge–Kutta-type integrator is introduced to treat the viscosity of the porous skeleton. The overall algorithm results in an efficient stabilized scheme allowing for linear equal interpolation of field variables, even when the so-called “undrained incompressible limit” is approached.The accuracy and stability of the method are verified with reference to a 1D benchmark problem, concerning the propagation of P waves along a saturated viscoelastic soil stratum. For this problem, a frequency-domain analytical solution is derived, assuming incompressible interstitial fluid and soil grains. The assumption of Maxwell viscoelastic soil skeleton is analytically convenient to preserve the linearity of the problem, while the same rheology of a more realistic elasto-viscoplastic non-linear behaviour is maintained.The performance of the fractional step Taylor–Galerkin algorithm is explored simulating the dynamic response of the stratum to harmonic, impulsive and seismic input excitations. In particular, parametric analyses are performed to confirm the effectiveness of the method in reproducing fully undrained responses, as well as in dealing with weakly viscous materials.     

Least-squares finite element formulation for shear-deformable shells

We present a least-squares based finite element formulation for the numerical analysis of shear-deformable shell structures. The variational problem is obtained by minimizing the least-squares functional, defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of Hilbert spaces. The use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise, i.e. stability requirements such as inf–sup conditions never arise. The proposed formulation retains the generalized displacements and stress resultants as independent variables and, in view of the nature of the variational setting upon which the finite element model is built, allows for equal-order interpolation. A p-type hierarchical basis is used to construct the discrete finite element model based on the least-squares formulation. Exponentially fast decay of the least-squares functional is verified for increasing order of the modal expansions. Several well established benchmark problems are solved to demonstrate the predictive capability of the least-squares based shell elements. Shell elements based on this formulation are shown to be effective in both membrane- and bending-dominated states.     

Isoparametric finite element analysis for solid rocket motors

Several large scale finite element computerprograms have been written for the stress analysis of solid propellant rocket motors during the past several years. The displacement formulation is employed, and continuum and shell elements are coupled together along the idealized interface of the grain or insulation and the motor case. However, since the continuum elements exhibit a linear variation in displacements along each edge, and the shell elements have a cubic variation in displacements along their length, gaps can develop at the common interface of a coupled continuum and shell element. This incompatibility may lead to errors in stress predictions at a very important area in the solid rocket motor, the case/liner or case/ grain bondline.

In this present paper, the problem of grain/case finite element incompatibility is addressed within the framework of the conventional displacement formulation. Both the shell elements and the continuum elements are derived from the same quadratic isoparametric shape function, yielding finite element idealizations which are totally compatible along the interface of a continuum and shell element. A numerical example is presented which shows the differences between the present compatible formulation and the conventional incompatible formulations.     

Large deformation static and dynamic finite element analysis of extensible cables

Approximate numerical integration of the element total potential energy with polynomial interpolation of the displacements creates high order nonlinear, extensible, cable finite elements. Successful computations of static and dynamic large displacement cable problems are carried out with the element.     

A discrete formulation for elastic solids with damaging interfaces

An elastic solid with embedded interfaces, loci of possible displacement discontinuities, is considered here. Decohesion and quasi-brittle fracture processes are simulated by making use of softening interface laws which relate tractions to displacements jumps. A discrete formulation in terms of interface variables only, is obtained. The space discretization is carried out by means of a mixed finite element approach in which all interface variables are modelled. Study of uniqueness of the rate problem formulated in terms of interface variables and of stability of the equilibrium states is presented. Some examples are shown in order to clarify the formulation.     

Galerkin finite element method for incompressible thermofluid flows framed within the bond graph theory

In this paper a bond graph methodology is used to model incompressible fluid flows with viscous and thermal effects. The distinctive characteristic of these flows is the role of pressure, which does not behave as a state variable but as a function that must act in such a way that the resulting velocity field has divergence zero. Velocity and entropy per unit volume are used as independent variables for a single-phase, single-component flow. Time-dependent nodal values and interpolation functions are introduced to represent the flow field, from which nodal vectors of velocity and entropy are defined as state variables. The system for momentum and continuity equations is coincident with the one obtained by using the Galerkin method for the weak formulation of the problem in finite elements. The integral incompressibility constraint is derived based on the integral conservation of mechanical energy. The weak formulation for thermal energy equation is modeled with true bond graph elements in terms of nodal vectors of temperature and entropy rates, resulting a Petrov–Galerkin method. The resulting bond graph shows the coupling between mechanical and thermal energy domains through the viscous dissipation term. All kind of boundary conditions are handled consistently and can be represented as generalized effort or flow sources. A procedure for causality assignment is derived for the resulting graph, satisfying the Second principle of Thermodynamics.     

A numerical study on the fluid compressibility effects in strongly coupled fluid–solid interaction problems

Interactions between an incompressible fluid passing through a flexible tube and the elastic wall is one of the strongly coupled fluid–solid interaction (FSI) problems frequently studied in the literature due to its research importance and wide range of applications. Although incompressible fluid is a prevalent model in many simulation studies, the assumption of incompressibility may not be appropriate in strongly coupled FSI problems. This paper narrowly aims to study the effect of the fluid compressibility on the wave propagation and fluid–solid interactions in a flexible tube. A partitioned FSI solver is used which employs a finite volume-based fluid solver. For the sake of comparison, both traditional incompressible (ico) and weakly compressible (wco) fluid models are used in an Arbitrary Lagrangian–Eulerian (ALE) formulation and a PISO-like algorithm is used to solve the unsteady flow equations on a collocated mesh. The solid part is modeled as a simple hyperelastic material obeying the St-Venant constitutive relation. Computational results show that not only use of the weakly compressible fluid model makes the FSI solver in this case more efficient, but also the incompressible fluid model may produce largely unrealistic computational results. Therefore, the use of the weakly compressible fluid model is suggested for strongly coupled FSI problems involving seemingly incompressible fluids such as water especially in cases where wave propagation in the solid plays an important role.

     

Galerkin-finite element formulation in viscous flow

A numerical solution procedure, based on the finite element idealization, combined with Galerkin's principle, has been derived for a class of problems in viscous fluid dynamics. In this note the authors present the results obtained for the simple case of boundary layer solution over a flat plate—the Blasius problem. The method shows excellent results with respect to other methods. Results emphasize the fact that better accuracy can be obtained by solving the non-linear equations without referring to linearization procedures. Also, the use of a few high order elements, can assure better accuracy than lower order elements.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.