Partitioned solvers for the fluid-structure interaction problems with a nearly incompressible elasticity model

In this paper, we present some analysis and numerical studies on two partitioned fluid-structure interaction solvers: a preconditioned GMRES solver and a Newton based solver, for the fluid-structure interaction problems employing a nearly incompressible elasticity model in a classical mixed displacement-pressure formulation. Both are highly relying on robust and efficient solvers for the fluid and structure sub-problems obtained from an extended and stabilized finite element discretization on hybrid meshes. A special algebraic multigrid method capable of handling such general saddle point systems for the incompressible and nearly incompressible models is investigated.     

Multi-solver algorithms for the partitioned simulation of fluid–structure interaction

In partitioned fluid–structure interaction simulations, the flow equations and the structural equations are solved separately. As a result, a coupling algorithm is needed to enforce the equilibrium on the fluid–structure interface in cases with strong interaction. This coupling algorithm performs coupling iterations between the solver of the flow equations and the solver of the structural equations. Current coupling algorithms couple one flow solver with one structural solver. Here, a new class of multi-solver quasi-Newton coupling algorithms for unsteady fluid–structure interaction simulations is presented. More than one flow solver and more than one structural solver are used for a single simulation. The numerical experiments demonstrate that the duration of a simulation decreases as the number of solvers is increased.     

A large scale parallel fluid-structure interaction computing platform for simulating structural responses to a detonation shock

Due to the intrinsic nature of multi-physics, it is prohibitively complex to design and implement a simulation software platform for study of structural responses to a detonation shock. In this article, a partitioned fluid-structure interaction computing platform is designed for parallel simulating structural responses to a detonation shock. The detonation and wave propagation are modeled in an open-source multi-component solver based on OpenFOAM and blastFoam, and the structural responses are simulated through the finite element library deal.II. To capture the interaction dynamics between the fluid and the structure, both solvers are adapted to preCICE. For improving the parallel performance of the computing platform, the inter-solver data is exchanged by peer-to-peer communications and the intermediate server in conventional multi-physics software is eliminated. Furthermore, the coupled solver with detonation support has been deployed on a computing cluster after considering the distributed data storage and load-balancing between solvers. The 3D numerical result of structural responses to a detonation shock is presented and analyzed. On 256 processor cores, the speedup ratio of the simulations for a detonation shock reach 178.0 with 5.1 million of mesh cells and the parallel efficiency achieve 69.5%. The results demonstrate good potential of massively parallel simulations. Overall, a general-purpose fluid-structure interaction software platform with detonation support is proposed by integrating open source codes. And this work has important practical significance for engineering application in fields of construction blasting, mining, and so forth.     

Two level algorithms for partitioned fluid-structure interaction computations

In this paper we use the multigrid algorithm - commonly used to improve the efficiency of the flow solver - to improve the efficiency of partitioned fluid-structure interaction iterations. Coupling not only the structure with the fine flow mesh, but also with the coarse flow mesh (often present due to the multigrid scheme) leads to a significant efficiency improvement. As solution of the flow equations typically takes much longer than the structure solve, and as multigrid is not standard in structure solvers, we do not coarsen the structure or the interface. As a result, the two level method can be easily implemented into existing solvers.Two types of two level algorithms were implemented: (1) coarse grid correction of the partitioning error and (2) coarse grid prediction or full multigrid to generate a better initial guess. The resulting schemes are combined with a fourth-order Runge-Kutta implicit time integration scheme. For the linear, one-dimensional piston problem with compressible flow the superior stability, accuracy and efficiency of the two level algorithms is shown. The parameters of the piston problem were chosen such that both a weak and a strong interaction case were obtained.Even the strong interaction case, with a flexible structure, could be solved with our new two level partitioned scheme with just one iteration on the fine grid. This is a major accomplishment as most weakly coupled methods fail in this case. Of the two algorithms the coarse grid prediction or full multigrid method was found to perform best. The resulting efficiency gain for our one-dimensional problem is around a factor of ten for the coarse to intermediate time steps at which the high-order time integration methods should be run. For two- and three-dimensional problems the efficiency gain is expected to be even larger.     

On comparative-static analysis in numerical nonlinear economic models

Many economic models can be formulated as a system of nonlinear equations and solved numerically using a black-box equation solver found in software packages such as EUREKA or MATHEMATICA. Unfortunately, black-box solvers can fail for several reasons and the messages displayed upon termination may not be very informative. In this paper we discuss a Jacobian-based solution method we devised for comparative static analysis of a model of parking in order to cope with computational difficulties encountered with black-box solvers. Our method worked well and should apply to a variety of computational economic (and non-economic) models.     

Performance of iterative methods in ANSYS on cray parallel/vector supercomputers

This paper describes recent work using iterative methods for the solution of linear systems in the ANSYS program. The ANSYS program, a general purpose finite element code widely used in structural analysis applications, has now added an iterative solver option. The development of robust iterative solvers and their use in commercial programs is discussed. Discussion of the applicability of iterative solvers as a general purpose solver will include the topics of robustness; as well as memory requirements and CPU performance. A new iterative solver for general purpose finite element codes which functions as a “black-box” solver using element-specific information and the underlying problem physics to construct an effective and inexpensive preconditioner is described. Some results are given from realistic examples comparing the performance of the iterative solver implemented in ANSYS with the traditional parallel/vector frontal solver used in ANSYS and a robust shifted incomplete Choleski iterative solver.     

A coupling algorithm for partitioned solvers applied to bubble and droplet dynamics

Bubbles and droplets both consist of a liquid in contact with a gas. In this paper, we consider the interface between the incompressible liquid and the gas as a zero thickness structure. The position of the interface is determined by the equilibrium between surface tension effects and the fluid pressure difference across the interface. So, the structure interacts with the fluids on either side. The behaviour of a limited number of bubbles and droplets can therefore be simulated as a Fluid-Structure Interaction (FSI) problem.Most existing techniques frequently used for studying bubble and droplet dynamics, such as Level Set or Volume Of Fluid, use monolithic schemes. The flow on both sides of the interface and the position of the interface are calculated in a single code. In this contribution, a partitioned approach is presented. The position of the interface is calculated with a structural solver. Given a displacement of the interface, a separate flow solver calculates the flow on the liquid side of the interface with the Arbitrary Lagrangian-Eulerian (ALE) technique. The structural solver uses a reduced order model of the flow solver to obtain implicit coupling between both solvers. This reduced order model is built up during the coupling iterations of a time step. Grid and time converged solutions of two axisymmetric problems are calculated: an oscillating water droplet in air and the growth and detachment of an air bubble from the outlet of a vertical needle, submerged in quiescent water.     

Performance of a new partitioned procedure versus a monolithic procedure in fluid–structure interaction

Fluid–structure interaction (FSI) can be simulated in a monolithic way by solving the flow and structural equations simultaneously and in a partitioned way with separate solvers for the flow equations and the structural equations. A partitioned quasi-Newton technique which solves the coupled problem through nonlinear equations corresponding to the interface position is presented and its performance is compared with a monolithic Newton algorithm. Various structural configurations with an incompressible fluid are solved, and the ratio of the time for the partitioned simulation, when convergence is reached, to the time for the monolithic simulation is found to be between 1/2 and 4. However, in this comparison of the partitioned and monolithic simulations, the flow and structural equations have been solved with a direct sparse solver in full Newton–Raphson iterations, only relatively small problems have been solved and this ratio would likely change if large industrial problems were considered or if other solution strategies were used.     

SMT求解器理论组合技术研究

可满足模理论(SMT)求解器是计算机科学中用来判定一阶逻辑公式可满足性的程序,是许多形式化方法的验证引擎.理论求解器实现了SMT基于不同理论背景的求解过程,然而实际问题常以多个理论为背景.因此,本文重点介绍理论组合判定方法,概述SMT求解器的发展现状,并分析了几个主流SMT求解器理论组合判定关键技术.通过对照实验,评估...     

Fluid–shell structure interaction analysis by coupled particle and finite element method

This paper presents a coupled particle and finite element method for fluid–shell structure interaction analysis. The Moving Particle Semi-Implicit (MPS) method is used to analyze fluid flow and the MITC4 shell element is used in the FEM analysis of the structure. This paper considers partitioned coupling between the fluid and structural solvers. In order to satisfy compatibility in the employed partitioned coupling scheme, the Neumann–Dirichlet condition is applied to both the fluid and the structure. A symplectic time integration scheme is used to preserve energy when analyzing the shell structure. If the frequencies of the shell analysis are much higher than those of the MPS fluid solver, its time integration scheme is sub-cycled. When the presented coupling scheme was applied to simulate the sloshing phenomenon in an elastic thin shell structure, fluid fragmentation and large structural deformations were observed.     

A Sparse Multiscale Algorithm for Dense Optimal Transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport, we provide a framework to verify global optimality of a discrete transport plan locally. This allows the construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multiscale scheme. Any existing discrete solver can be used as internal black-box. We explicitly describe how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance. Significant reductions in run-time and memory requirements have been observed.     

A parsimony tree for the SAT2002 competition

Twenty of the programs (solvers) submitted to the SAT 2002 Contest had no disqualifying errors. These solvers were run on 2023 satisfiability problems of varying hardnesses. Each solver was judged by which problems it could solve within the allowed time limit. Twelve solvers were best on some problem — they could solve it and the others could not. Only two solvers could not beat each remaining solver on some problems (where the problems could vary depending on which solver it was trying to beat). Thus, there is evidence that 18 solvers were extremely good. It is interesting to analyze the contest results in a way that groups together solvers with similar strengths and weaknesses. This paper uses the parsimony algorithm to produce a classification of the twenty solvers. The paper also has a second classification, almost the same as the first, for the twenty solvers, updated versions of two solvers, and a fictitious state of the art solver. The contest problems came in three groups, Industrial, Hand Made, and Random. The Random group of problems was about three times as large as the other two together. The classification identifies four groups of solvers (plus a miscellaneous group): weak solvers, incomplete solvers which are very good at some satisfiable Random problems, complete solvers which are very good at most Random problems, and complete solvers which are very good at Industrial and Hand Made problems.     

A parsimony tree for the SAT2002 competition

Twenty of the programs (solvers) submitted to the SAT 2002 Contest had no disqualifying errors. These solvers were run on 2023 satisfiability problems of varying hardnesses. Each solver was judged by which problems it could solve within the allowed time limit. Twelve solvers were best on some problem – they could solve it and the others could not. Only two solvers could not beat each remaining solver on some problems (where the problems could vary depending on which solver it was trying to beat). Thus, there is evidence that 18 solvers were extremely good. It is interesting to analyze the contest results in a way that groups together solvers with similar strengths and weaknesses. This paper uses the parsimony algorithm to produce a classification of the twenty solvers. The paper also has a second classification, almost the same as the first, for the twenty solvers, updated versions of two solvers, and a fictitious state of the art solver. The contest problems came in three groups, Industrial, Hand Made, and Random. The Random group of problems was about three times as large as the other two together. The classification identifies four groups of solvers (plus a miscellaneous group): weak solvers, incomplete solvers which are very good at some satisfiable Random problems, complete solvers which are very good at most Random problems, and complete solvers which are very good at Industrial and Hand Made problems.     

Energy conservation under incompatibility for fluid-structure interaction problems

Concurrent numerical methods for fluid-structure interaction problems are typically based on partitioned solution procedures. However, such partitioned methods are inherently non-conservative. In the present work, we investigate the conservation properties of monolithic discretisations for fluid-structure interaction problems. We consider a prototypical fluid-structure interaction problem, viz., the piston problem. A variational formulation allows us to establish precisely the conservation properties of the continuum problem and its discretisation by the finite-element method. We show that the conservation of energy by monolithic discretisations is only trivially maintained under restrictive compatibility conditions on the approximation spaces in the fluid and the structure. Moreover, we introduce a new discretisation based on coincidence conditions which ensures energy conservation under incompatibility. Numerical results which illustrate the effectiveness of the new discretisation are presented.     

Black-box decomposition approach for computational hemodynamics: One-dimensional models

Increasing efforts exist in integrating different levels of detail in models of the cardiovascular system. For instance, one-dimensional representations are employed to model the systemic circulation. In this context, effective and black-box-type decomposition strategies for one-dimensional networks are needed, so as to: (i) employ domain decomposition strategies for large systemic models (1D–1D coupling) and (ii) provide the conceptual basis for dimensionally-heterogeneous representations (1D–3D coupling, among various possibilities). The strategy proposed in this article works for both of these two scenarios, though the several applications shown to illustrate its performance focus on the 1D–1D coupling case.A one-dimensional network is decomposed in such a way that each coupling point connects two (and not more) of the sub-networks. At each of the M connection points two unknowns are defined: the flow rate and pressure. These 2M unknowns are determined by 2M equations, since each sub-network provides one (non-linear) equation per coupling point. It is shown how to build the 2M × 2M non-linear system with arbitrary and independent choice of boundary conditions for each of the sub-networks. The idea is then to solve this non-linear system until convergence, which guarantees strong coupling of the complete network. In other words, if the non-linear solver converges at each time step, the solution coincides with what would be obtained by monolithically modeling the whole network. The decomposition thus imposes no stability restriction on the choice of the time step size.Effective iterative strategies for the non-linear system that preserve the black-box character of the decomposition are then explored. Several variants of matrix-free Broyden’s and Newton-GMRES algorithms are assessed as numerical solvers by comparing their performance on sub-critical wave propagation problems which range from academic test cases to realistic cardiovascular applications. A specific variant of Broyden’s algorithm is identified and recommended on the basis of its computer cost and reliability.     

Randomized constraint solvers: a comparative study

The complexity of constraints is a major obstacle for constraint-based software verification. Automatic constraint solvers are fundamentally incomplete: input constraints often build on some undecidable theory or some theory the solver does not support. This paper proposes and evaluates several randomized solvers to address this issue. We compared the effectiveness of a symbolic solver (CVC3), a random solver, two heuristic search solvers, and seven hybrid solvers (i.e. mix of random, symbolic, and heuristic solvers). We evaluated the solvers on a benchmark generated with a concolic execution of 9 subjects. The performance of each solver was measured by its precision, which is the fraction of constraints that the solver can find solution out of the total number of constraints that some solver can find solution. As expected, symbolic solving subsumes the other approaches for the 4 subjects that only generate decidable constraints. For the remaining 5 subjects, which contain undecidable constraints, the hybrid solvers achieved the highest precision (fraction of constraints that a solver can find a solution out of the total number of satisfiable constraints). We also observed that the solvers were complementary, which suggests that one should alternate their use in iterations of a concolic execution driver.     

VERICOMP: a system to compare and assess verified IVP solvers

Obtaining verified numerical solutions to initial value problems (IVPs) for ordinary differential equations (ODEs) is important in many application areas (e.g. biomechanics or automatic control). During the last decades, a number of solvers have been developed for this purpose. However, they are rarely used by industry engineers. One reason for this is the lack of information about what tool with what settings to choose. Therefore, it is necessary to develop a system for testing the available tools and recommending an ODE solver best suited for the task at hand in the area of verified software. In this paper, we present the first version of our web-based platform VERICOMP for assessing verified IVP solvers. We discuss a possible classification for user problems, suitable comparison criteria and measures for the quantification of overestimation. After that, we introduce the platform itself, which allows us to compare different solvers on a problem class or to evaluate the performance of a single solver on different problem classes. In addition, we describe how to extend VERICOMP with a recommender system that automatically suggests the most suitable solver based on user preferences and solver statistics.     

基于INTESIM-GISCI的流固耦合 仿真软件技术及应用

为突破传统商业软件流固耦合分析仅局限于内部预先集成的流体和结构求解器的约束,使流固耦合分析更具开放性和可拓展性,以耦合驱动程序INTESIM GISCI为框架体系,在开源流体求解器SU2的源代码上加入时间同步点和相关函数功能形成INTESIM-SU2,使INTESIM-SU2可以与原有的结构求解器INTESIM-Structure通过耦合界面传递数据的方式进行耦合分析,从而实现基于耦合驱动程序INTESIM-GISCI的流固耦合仿真软件开发。将多个流固耦合分析案例与其他商业软件进行对比,证明基于INTESIM-GISCI的流固耦合仿真软件可广泛应用于实际工程问题的仿真分析。     

Parallel methods for optimality criteria-based topology optimization

Topology optimization problems require the repeated solution of finite element problems that are often extremely ill-conditioned due to highly heterogeneous material distributions. This makes the use of iterative linear solvers inefficient unless appropriate preconditioning is used. Even then, the solution time for topology optimization problems is typically very high. These problems are addressed by considering the use of non-overlapping domain decomposition-based parallel methods for the solution of topology optimization problems. The parallel algorithms presented here are based on the solid isotropic material with penalization (SIMP) formulation of the topology optimization problem and use the optimality criteria method for iterative optimization. We consider three parallel linear solvers to solve the equilibrium problem at each step of the iterative optimization procedure. These include two preconditioned conjugate gradient (PCG) methods: one using a diagonal preconditioner and one using an incomplete LU factorization preconditioner with a drop tolerance. A third substructuring solver that employs a hybrid of direct and iterative (PCG) techniques is also studied. This solver is found to be the most effective of the three solvers studied, both in terms of parallel efficiency and in terms of its ability to mitigate the effects of ill-conditioning. In addition to examining parallel linear solvers, we consider the parallelization of the iterative optimality criteria method. To tackle checkerboarding and mesh dependence, we propose a multi-pass filtering technique that limits the number of “ghost” elements that need to be exchanged across interprocessor boundaries.     

Comparing direct and iterative equation solvers in a large structural analysis software system

This paper describes two vectorized implementations of preconditioned conjugate gradient (PCG) solvers. Sparse and diagonal matrix storage schemes are described and compared. A vectorized incomplete Choleski preconditioning is described and compared with Jacobi preconditioning. A modification to the basic no-fill incomplete Choleski method to improve performance and robustness is described. The two PCG solvers are compared with direct Choleski methods using a sparse Choleski solver from SPARSPAK and a vectorized variable-band Choleski solver developed at NASA Langley Research Center. All of the linear equation solvers are implemented in a large structural analysis finite element software system called the Computational Structural Mechanics (CSM) Testbed. The CSM Testbed is used to provide a common software system in which new methods are developed and tested. Several representative two- and three-dimensional structural analysis problems are solved using the various equation solvers. Results are given from runs made on the CONVEX C220 and CRAY 2 computer systems. Comparisons of the convergence rates for the iterative solvers as well as the computation rates, number of operations, and overall CPU time required by all of the equation solvers are given.