Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.     

A coupled PFEM–Eulerian approach for the solution of porous FSI problems

This paper aims to present a coupled solution strategy for the problem of seepage through a rockfill dam taking into account the free-surface flow within the solid as well as in its vicinity. A combination of a Lagrangian model for the structural behavior and an Eulerian approach for the fluid is used. The particle finite element method is adopted for the evaluation of the structural response, whereas an Eulerian fixed-mesh approach is employed for the fluid. The free surface is tracked by the use of a level set technique. The numerical results are validated with experiments on scale models rockfill dams.     

Automatic monitoring of element shape quality in 2-D and 3-D computational mesh dynamics

One of the major problems in fluid–structure interaction using the arbitrary Lagrangian Eulerian approach lies in the area of dynamic mesh generation. For accurate fluid-dynamic computations, meshes must be generated at each time step. The fluid mesh must be regenerated in the deformed fluid domain in order to account for the displacements of the elastic body computed by the structural dynamics solver. In the elasticity-based computational dynamic mesh procedure, the fluid mesh is modeled as a pseudo-elastic solid the deformation of which is based on the displacement boundary conditions, resulting from the solution of the computational structural dynamics problem. This approach has a distinct advantage over other mesh-movement algorithms in that it is a very general, physically based approach that can be applied to both structured and unstructured meshes. The major drawback of the linear elastostatic solver is that it does not guarantee the absence of severe element distortion. This paper describes a novel mesh-movement procedure for mesh quality control of 2-D and 3-D dynamic meshes based on solving a pseudo-nonlinear elastostatic problem. An inexpensive distortion measure for different types of elements is introduced and used for controlling the element shape quality. The mesh-movement procedure is illustrated with several examples (large-displacement and free-boundary problems) that highlight its advantages in terms of performance, mesh quality, and robustness. It is believed that the resulting scheme will result in a more economical simulation of the motion of complex geometry, 3-D elastic bodies immersed in temporally and spatially evolving flows. Received 20 April 2000     

Analysis of inequality and residual flow procedures and an iterative scheme for free surface seepage

An efficient block overrelaxation algorithm is presented for the solution of the steady-state dam seepage problem. The formulation of Alt is used in conjunction with a finite element method on a fixed mesh to obtain a finite dimensional problem which is then solved by the algorithm. Some similarities of the formulation and the residual flow method of Desai are also discussed. Numerical results are compared with results of other authors.     

Non‐linear seismic behaviour of concrete gravity dams using coupled finite element–boundary element technique

In this paper, the seismic response of concrete gravity dams is presented using the concept of Continuum Damage Mechanics (CDM) and adopting the hybrid Finite Element–Boundary Element technique (FE–BE). The finite element method is used for discretization of the near field and the boundary element method is employed to model the semi‐infinite far field. Because of the non‐linear nature of the discretizied equations of motion modified Newton–Raphson approach has been used at each time step to linearize them. Damage evolution based on tensile principal strain using mesh‐dependent hardening modulus technique is adopted to ensure the mesh objectivity and to calculate the accumulated damage. The methodology employed is shown to be computationally efficient and consistent in its treatment of both damage growth and damage propagation in gravity dams. Other important features considered in the analysis are: (1) realistic damage modelling for concrete that allows isotropic as well as anisotropic damage state and exhibits stiffness recovery upon load reversals. (2) softening initiation and strain softening criteria for concrete, and (3) proper modelling of semi‐infinite foundation using FE–BE method that allows to consider dam–foundation interaction analysis. As an application of the proposed formulation a gravity dam has been analysed and the results are compared with different foundation stiffnesses. The results of the analysis indicate the importance of including rock foundation in the seismic analysis of dams. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive error analysis in seepage problems

Numerical experiments in adapting variations of a computationally simple error estimator (the Zienkiewicz-Zhu estimator) to an existing finite element code are shown. The error estimator used allows both overall and local errors to be estimated. From the local estimates of error, refinements of the mesh are calculated to reach a prescribed error tolerance. These calculated refinements are used by a mesh refiner to produce a modified mesh which lowers the overall error to the prescribed value while keeping the mesh as crude as possible. The physical example on which these numerical experiments are performed is that of free surface flow through an earth dam with a toe drain. It is also shown how the problem formulation affects the error analysis and how the choice of computational scheme affects the mesh adaptation.     

Seismic analysis of arch dams—a continuum damage mechanics approach

In this paper, the non‐linear seismic response of arch dams is presented using the concept of Continuum Damage Mechanics (CDM). The analysis is performed using the finite element technique and appropriate non‐linear material and damage models in conjunction with the α‐algorithm for time marching. Because of the non‐linear nature of the discretizied equations of motion, modified Newton–Raphson approach has been used at each time step. Damage evolution based on tensile principal strain using mesh‐dependent hardening modulus technique is adopted to ensure the mesh objectivity and to calculate the accumulated damage. The methodology employed is shown to be computationally efficient and consistent in its treatment of both damage growth and damage propagation. As an application of the proposed formulation, a double curvature arch dam has been analysed and the results are compared with the solutions from linear analysis and it is shown that the structural response of arch dams varies significantly in terms of damage evolution. Copyright © 1999 John Wiley & Sons, Ltd.     

Accelerated and robust finite volume Navier-Stokes solver for all speeds

The paper describes the results obtained from a multigrid accelerated Navier-Stokes solver. The method is based on 2-D explicit cell-centred finite volume Reynolds averaged Navier-Stokes (N-S) flow solver for speeds from near-incompressible Mach numbers to high hypersonic Mach numbers including flows at high angles of attack. The time integration is done using a hybrid 5-stage Runge-Kutta local time stepping scheme. With the help of a simple technique, the capability of the Jameson-Schmidt-Turkel numerical dissipation scheme has been enhanced to include hypersonic flows. The iterative procedure is accelerated significantly by incorporating a multigrid technique which has been used in all computations up to about supersonic speeds. Systematic numerical experiments were conducted to evolve guidelines to generate airfoil grid that could offer reliable flow simulations. The computed results are in very good agreement with experimental data where available, especially from the point of view of predicting large suction peaks and shock locations where considerable departures are often seen in the literature. Further, the highly accelerated computations make this code a useful tool of practical interest in preliminary aerodynamic design.     

R-ADAPTIVE BOUNDARY ELEMENT METHOD FOR UNSTEADY FREE-SURFACE FLOW ANALYSIS

An adaptive meshing based on the r-method is developed for two-dimensional unsteady non-linear flows with a free surface. Coupling of a boundary element equation and a weighted residual formulation of the pressure equation on the free surface is employed in solving the wave problems. A mesh optimization scheme is constructed for these two matrix equations. The final mesh distribution on the free surface is determined from the weighted average of these two adaptive meshes. Through numerical analyses of a non-linear sloshing problem and solitary waves in a tank, the influence of time interval of the remeshing and the weight factor for two adaptive meshes on the accuracy and efficiency of the proposed approach is investigated.     

Solution of non-linear boundary value problems by discrete least squares

This paper presents a new discrete (point-matching) least squares method for solving general, non-linear boundary value problems. The technique uses a mixed formulation and minimizes the sum of squared residuals by Powell's (1965) algorithm. The present method is compared with finite difference and weighted residual finite element methods in the examples and discussion. It shows considerable promise as a general differential equation solver for both large and small computer systems.     

Modeling the cracks opening–closing and possible remedial sawing operation of AAR-affected dams

A significant number of existing concrete dams are at present deteriorated by chemo-mechanical processes known as Alkali-Aggregate Reactions (AAR). This phenomenon is responsible for cracking and expansion of the material which lead to mechanical degradation of the structures operability and safety. Its chemo-mechanical modeling has been already dealt with by considering the influence of temperature, humidity, stiffness reduction and stress in the development of expansion. This modeling has first been developed as valid only for continuous media. In order to widen the field of modeling applicability, this paper describes a novel procedure for modeling displacement discontinuities in AAR-affected concrete structures, mainly to take into account two phenomena: firstly, the modeling of significant isolated cracks opening–closing that result from interactions between expansion and mechanical behavior of the structure; and secondly, the evaluation of the potential efficiency of stress release (e.g. sawing of dams), the procedure of which consists in the creation of slot cutting in the structure for release of excessive compressive stresses. The elaborated model, aimed to possibly combine contact elements and non-linear volume coupled models of durability mechanics related to AAR modeling, was applied to the 3D simulations of a simplified model ideally representing a part of a dam and also to a real gravity dam. Displacements and stresses at the contact elements zone were computed and compared with results of the dam computations without contact elements. The results show that the creation of the slot cutting leads to decrease of the compressive stresses in the structure, which confirms a favorable effect of this stress release technique in order to deal with AAR affected structures.     

Unified fluid–structure interpolation and mesh motion using radial basis functions

A multivariate interpolation scheme, using radial basis functions, is presented, which results in a completely unified formulation for the fluid–structure interpolation and mesh motion problems. The method has several significant advantages. Primarily, all volume mesh, structural mesh, and flow‐solver type dependence is removed, and all operations are performed on totally arbitrary point clouds of any form. Hence, all connectivity and user‐input requirements are removed from the computational fluid dynamics–computational structural dynamics (CFD–CSD) coupling problem, as only point clouds are required to determine the coupling. Also, it may equally well be applied to structured and unstructured grids, or structural and aerodynamic grids that intersect, again because no connectivity information is required. Furthermore, no expensive computations are required during an unsteady simulation, just matrix–vector multiplications, since the required dependence relations are computed only once prior to any simulation and then remain constant. This property means that the method is both perfectly parallel, since only the data relevant to each structured block or unstructured partition are required to move those points, and totally independent from the flow solver. Hence, a completely generic ‘black box’ tool can be developed, which is ideal for use in an optimization approach. Aeroelastic behaviour of the Brite–Euram MDO wing is analysed in terms of both static deflection and dynamic responses, and it is demonstrated that responses are strongly dependent on the exact CFD–CSD interpolation used. Mesh quality is also examined during the motion resulting from a large surface deformation. Global grid quality is shown to be preserved well, with local grid orthogonality also being maintained well, particularly at and near the moving surface, where the original orthogonality is retained. Copyright © 2007 John Wiley & Sons, Ltd.     

二维堤坝管涌的数值模拟研究

根据修正的SPV(Stavropoulou-Papanastasiou-Vardoulakis)管涌控制方程,采用Newton-Raphson求解法和传导矩阵调整法,对二维带自由面的均质土石坝坝体渗流侵蚀问题进行了有限元数值分析。数值分析表明:在具有渗流自由面的均质土石坝坝体中,溢出面和自由面附近区域的水力梯度较大,比其他区域更容易大于管涌临界水力梯度。管涌侵蚀区(孔隙率明显增大)和细颗粒最大浓度区首先出现在溢出面处,然后沿着渗流自由面附近区域向上游扩展,最终达到自由面附近的进水面,形成孔隙率明显增大的侵蚀贯通通道。管涌侵蚀增大堤坝的渗透系数,它对堤坝渗透速度的影响较大,但对孔压分布和自由面位置的影响较小。     

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

This paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities. The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision. This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.     

Numerical simulations of strain localization in inelastic solids using mesh‐free methods

In this paper, a comprehensive account on using mesh‐free methods to simulate strain localization in inelastic solids is presented. Using an explicit displacement‐based formulation in mesh‐free computations, high‐resolution shear‐band formations are obtained in both two‐dimensional (2‐D) and three‐dimensional (3‐D) simulations without recourse to any mixed formulation, discontinuous/incompatible element or special mesh design. The numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi‐uniform, which, to a large extent, relieves the mesh alignment sensitivity that finite element methods suffer. Moreover, a simple h‐adaptivity procedure is implemented in the explicit calculation, and by utilizing a mesh‐free hierarchical partition of unity a spectral (wavelet) adaptivity procedure is developed to seek high‐resolution shear‐band formations. Moreover, the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine‐scale finite element simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Three dimensional automatic refinement method for transient small strain elastoplastic finite element computations

In this paper, the refinement strategy based on the “Non-Linear Localized Full MultiGrid” solver originally published in Int. J. Numer. Meth. Engng 84(8):947–971 (2010) for 2-D structural problems is extended to 3-D simulations. In this context, some extra information concerning the refinement strategy and the behavior of the error indicators are given. The adaptive strategy is dedicated to the accurate modeling of elastoplastic materials with isotropic hardening in transient dynamics. A multigrid solver with local mesh refinement is used to reduce the amount of computational work needed to achieve an accurate calculation at each time step. The locally refined grids are automatically constructed, depending on the user prescribed accuracy. The discretization error is estimated by a dedicated error indicator within the multigrid method. In contrast to other adaptive procedures, where grids are erased when new ones are generated, the previous solutions are used recursively to reduce the computing time on the new mesh. Moreover, the adaptive strategy needs no costly coarsening method as the mesh is reassessed at each time step. The multigrid strategy improves the convergence rate of the non-linear solver while ensuring the information transfer between the different meshes. It accounts for the influence of localized non-linearities on the whole structure. All the steps needed to achieve the adaptive strategy are automatically performed within the solver such that the calculation does not depend on user experience. This paper presents three-dimensional results using the adaptive multigrid strategy on elastoplastic structures in transient dynamics and in a linear geometrical framework. Isoparametric cubic elements with energy and plastic work error indicators are used during the calculation.     

A perturbation analysis and finite element approximate model for free surface flow over curved beds

A perturbation analysis is developed for 1-D shallow water flow over a curved bed for applications such as spillways. The perturbation approach leads to a new formulation of the problem with associated weak integral statement and approximation using finite elements. The flow may exhibit a hydraulic jump in the downstream regime. An artificial dissipation technique is introduced to stabilize the non-linear problem and suppress numerical oscillations. Numerical results demonstrate the performance of the model and compare it with the steep-slope shallow water formulation corresponding to the model with zero curvature.     

浅谈降低深孔梯段爆破大块率的措施

面板堆石坝是当今世界发展的主要坝型之一。为节省投资,近年来利用枢纽建筑物的开挖弃料进行坝体石碴填筑,因此对控制爆破产出的石碴级配料、尤其是大块率提出严格要求。通过西大洋水库正常溢洪道深孔梯段爆破的大量工程实践,从施工工艺、爆破参数、工程地质等方面分析了爆破试验大块率偏高的原因,并提出了降低大块率的具体措施。实践证明,只要采取行之有效的技术措施,完全可以控制大块率。     

3D domain decomposition for violent wave‐ship interactions

A 3D Domain‐Decomposition (DD) strategy has been developed to deal with violent wave‐ship interactions involving water‐on‐deck and slamming occurrence. It couples a linear potential flow seakeeping solver with a Navier–Stokes method. The latter is applied in an inner domain where slamming, water‐on‐deck, and free surface fragmentation may occur, involving important flow nonlinearities. The field solver combines an approximated projection method with a level set technique for the free surface evolution. A hybrid strategy, combining the Eulerian level set concept to Lagrangian markers, is used to enforce more accurately the body boundary condition in case of high local curvatures. Main features of the weak and the strong coupling algorithms are described with special focus on the boundary conditions for the inner solver. Two ways of estimating the nonlinear loads by the Navier–Stokes method are investigated, on the basis of an extrapolation technique and an interpolation marching cubes algorithm, respectively. The DD is applied for the case of a freely floating patrol ship in head sea regular waves and compared against water‐on‐deck experiments in terms of flow evolution, body motions, and pressure on the hull. Improvement of the solver efficiency and accuracy is suggested. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids

This paper presents the development and validation of a parallel unstructured‐grid fluid–structure interaction (FSI) solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries. The Navier–Stokes solver on unstructured moving grid using the arbitrary Lagrangian Eulerian formulation is based on the artificial compressibility approach and a high‐order characteristics‐based finite‐volume scheme. Both unsteady flow and FSI are calculated with a matrix‐free implicit dual time‐stepping scheme. A membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions. This model can be employed to calculate arbitrary wall movement and variable tension along the membrane, together with a dynamic mesh method for large deformation of the flow field. The parallelization of the fluid–structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data. The parallel solver is used to simulate fluid flow in a two‐dimensional channel with and without moving membrane for validation and performance evaluation purposes. The speedups and parallel efficiencies obtained by this method are excellent, using up to 16 processors on a SGI Origin 2000 parallel computer. A maximum speedup of 23.14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver. The parallel results obtained are compared with those using serial code and they are found to be identical. Copyright © 2005 John Wiley & Sons, Ltd.