Reduced-dissipation remapping of velocity in staggered arbitrary Lagrangian-Eulerian methods

Remapping is an essential part of most Arbitrary Lagrangian-Eulerian (ALE) methods. In this paper, we focus on the part of the remapping algorithm that performs the interpolation of the fluid velocity field from the Lagrangian to the rezoned computational mesh in the context of a staggered discretization. Standard remapping algorithms generate a discrepancy between the remapped kinetic energy, and the kinetic energy that is obtained from the remapped nodal velocities which conserves momentum. In most ALE codes, this discrepancy is redistributed to the internal energy of adjacent computational cells which allows for the conservation of total energy. This approach can introduce oscillations in the internal energy field, which may not be acceptable. We analyze the approach introduced in Bailey (1984) [11] which is not supposed to introduce dissipation. On a simple example, we demonstrate a situation in which this approach fails. A modification of this approach is described, which eliminates (when it is possible) or reduces the energy discrepancy.     

R-Adaptive Reconnection-Based Arbitrary Lagrangian Eulerian Method-R-ReALE

We present a new R-adaptive Arbitrary Lagrangian Eulerian (ALE) method, based on the reconnection-based ALE - ReALE methodology [5, 41, 42]. The main elements in a standard ReALE method are: an explicit Lagrangian phase on an arbitrary polygonal (in 2D) mesh, followed by a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred onto the new grid. The rezoned mesh is smoothed by using one or several steps toward centroidal Voronoi tessellation, but it is not adapted to the solution in any way. We present a new R-adaptive ReALE method (R-ReALE, where R stands for Relocation). The new method is based on the following design principles. First, a monitor function (or error indicator) based on Hessian of some flow parameter(s), is utilized. Second, the new algorithm uses the equidistribution principle with respect to the monitor function as criterion for defining an adaptive mesh. Third, centroidal Voronoi tessellation is used for the construction of the adaptive mesh. Fourth, we modify the raw monitor function (scale it to avoid extremely small and large cells and smooth it to create a smooth mesh), in order to utilize theoretical results related to centroidal Voronoi tessellation. In the R-ReALE method, the number of mesh cells is chosen at the beginning of the calculation and does not change with time, but the mesh is adapted according to the modified monitor function during the rezone stage at each time step. We present all details required for implementation of the new adaptive R-ReALE method and demonstrate its performance relative to standard ReALE method on a series of numerical examples.     

The error‐minimization‐based rezone strategy for arbitrary Lagrangian‐Eulerian methods

The objective of the Arbitrary Lagrangian‐Eulerian (ALE) methodology for solving multidimensional fluid flow problems is to move the computational mesh, using the flow as a guide, to improve the robustness, accuracy and efficiency of a simulation. The main elements in the ALE simulation are an explicit Lagrangian phase, a rezone phase in which a new mesh is defined, and a remapping (conservative interpolation) phase, in which the Lagrangian solution is transferred to the new mesh. In most ALE codes, the main goal of the rezone phase is to maintain high quality of the rezoned mesh. In this article, we describe a new rezone strategy which minimizes the L₂ norm of the solution error and maintains smoothness of the mesh. The efficiency of the new method is demonstrated with numerical experiments. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Mesh rezoning for a dynamic triangular finite-elements hydrocode

A Lagrangian type of finite element hydrocode has one major disadvantage as it cannot handle severely distorted elements. A remedy to this problem is rezoning. In most works concerning rezoning, the entire domain simulated is rezoned at times determined by input data. In most problems rezoning is necessary only at certain times and in certain local regions. This paper presents a new approach: the time of rezoning is automatically determined, and only local regions are rezoned when necessary. The new mesh is generated by automatic triangulation. The assignment of values of variables to the nodes and elements of the new mesh is done in such a way that mass, internal energy, momentum, and kinetic energy are conserved for the region rezoned. The approach was programmed into the Lagrangian hydrocode DEFEL and penetration problems were simulated as examples. Even though the penetration depth is in agreement with the result from an approximate erosion algorithm, a more accurate shape of the penetrator tip was obtained by the new rezoning approach.     

Second-order negative-curvature methods for box-constrained and general constrained optimization

A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.     

A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY

This paper presents a second-order direct arbitrary Lagrangian Eulerian (ALE) method for compressible flow in two-dimensional cylindrical geometry.This algorithm has half-face fluxes and a nodal velocity solver,which can ensure the compatibility between edge fluxes and the nodal flow intrinsically.In two-dimensional cylindrical geometry,the control vol-ume scheme and the area-weighted scheme are used respectively,which are distinguished by the discretizations for the source term in the momentum equation.The two-dimensional second-order extensions of these schemes are constructed by employing the monotone up-wind scheme of conservation law (MUSCL) on unstructured meshes.Numerical results are provided to assess the robustness and accuracy of these new schemes.     

Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016     

求解流固耦合问题的一种四步分裂有限元算法

基于arbitrary Lagrangian Eulerian (ALE) 有限元方法,发展了一种求解流固耦合问题的弱耦合算法．将半隐式四步分裂有限元格式推广至求解ALE描述下的Navier-Stokes（N-S）方程,并在动量方程中引入迎风流线（streamline upwind/Petrov-Galerkin, SUPG）稳定项以消除对流引发的速度场数值振荡;采用Newmark-β法对结构方程进行时间离散;运用经典的Galerkin有限元法求解修正的Laplace方程以实现网格更新,每个计算步施加网格总变形量防止结构长时间、大位移运动时的网格质量恶化．运用上述算法对弹性支撑刚性圆柱体的流致振动问题进行了数值模拟,计算结果与已有结果相吻合,初步验证了该算法的正确性和有效性．     

Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints

In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original MPCC are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of MPCC. Numerical experiments are done to test the computational performances of several methods for MPCC proposed in the literature. This research was partially supported by the Research Grants Council (BQ654) of Hong Kong and the Postdoctoral Fellowship of The Hong Kong Polytechnic University. Dedicated to Alex Rubinov on the occassion of his 65th birthday.     

地形构造中地震波传播的非对称交错网格模拟

提出了一种新的三维空间对称交错网格差分方法,模拟地形构造中弹性波传播过程．通过具有二阶时间精度和四阶空间精度的不规则网格差分算子用来近似一阶弹性波动方程,引入附加差分公式解决非均匀交错网格的不对称问题．该方法无需在精细网格和粗糙网格间进行插值,所有网格点上的计算在同一次空间迭代中完成．使用精细不规则网格处理海底粗糙界面、 断层和空间界面等复杂几何构造, 理论分析和数值算例表明, 该方法不但节省了大量内存和计算时间, 而且具有令人满意的稳定性和精度．在模拟地形构造中地震波传播时,该方法比常规方法效率更高．     

Time-discrete higher order ALE formulations: a priori error analysis

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection–diffusion model defined on moving domains and written in the Arbitrary Lagrangian Eulerian (ALE) framework. Our estimates hold without any restrictions on the time steps for dG with exact integration or Reynolds’ quadrature. They involve a mild restriction on the time steps for the practical Runge–Kutta–Radau methods of any order. The key ingredients are the stability results shown earlier in Bonito et al. (Time-discrete higher order ALE formulations: stability, 2013) along with a novel ALE projection. Numerical experiments illustrate and complement our theoretical results.     

Iterative method for computing the Moore-Penrose inverse based on Penrose equations

An iterative algorithm for estimating the Moore-Penrose generalized inverse is developed. The main motive for the construction of the algorithm is simultaneous usage of Penrose equations (2) and (4). Convergence properties of the introduced method as well as their first-order and second-order error terms are considered. Numerical experiment is also presented.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).     

Formulation and survey of ALE method in nonlinear solid mechanics

This paper investigates the applicability and accuracy of existing formulation methods in general purpose finite element programs to the finite strain deformation problems. The basic shortcomings in using such programs in these applications are then pointed out and the need for a different type of formulation is discussed. An arbitrary Lagrangian-Eulerian (ALE) method is proposed and a concise survey of ALE formulation is given. A consistent and complete ALE formulation is derived from the virtual work equation transformed to arbitrary computational reference configurations. Differences between the proposed formulations and similar ones in the literature are discussed. The proposed formulation presents a general approach to ALE method. It includes load correction terms and is suitable for rate-dependent and rate-independent material constitutive law. The proposed formulation reduces to both updated Lagrangian and Eulerian formulations as special cases.     

A hybrid level-set/moving-mesh interface tracking method

An approach for combining Arbitrary–Lagrangian–Eulerian (ALE) moving-mesh and level-set interface tracking methods is presented that allows the two methods to be used in different spatial regions and coupled across the region boundaries. The coupling allows interface shapes to be convected from the ALE method to the level-set method and vice-versa across the ALE/level-set boundary. The motivation for this is to allow high-order ALE methods to represent interface motion in regions where there is no topology change, and the level-set function to be used in regions where topology change occurs. The coupling method is based on the characteristic directions of information propagation and can be implemented in any geometrical configuration. In addition, an iterative method for the hybrid formulation has been developed that can be combined with pre-existing solution methods. Tests of a propagating interface in a uniform flow show that the hybrid approach provides accuracy equivalent to what one is able to obtain with either of the methods individually.     

Augmentability in optimization theory

The present paper is concerned with the concept of augmentability. One form of augmentability is based on the generalized Lagrangian and leads to the method of multipliers for solving constrained minimum problems. An assumption of augmentability can be used in place of regularity to establish first-order and second-order necessary conditions for a minimum for constrained minimum problems. Another type of augmentability leads to the theory of Mayer fields in variational theory.Dedicated to R. Bellman     

Central schemes for Ideal magnetohydrodynamics

We present second-order accurate central finite volume methods adapted to three-dimensional ideal magnetohydrodynamics problems. These methods alternate between two staggered grids, thus leading to Riemann solver-free algorithms with relatively favorable computing times. The div· B = 0 constraint on the magnetic field is enforced with a suitable adaptation of the constrained transport method to our central schemes. Numerical experiments show the feasibility of the proposed methods and our results are in good agreement with existing results in the recent literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Seeking (and finding) Lagrangians

It is well known that for any second-order ordinary differential equation (ODE), a Lagrangian always exists, and the key to its construction is the Jacobi last multiplier. Is it possible to find Lagrangians for first-order systems of ODEs or for higher-order ODEs? We show that the Jacobi last multiplier can also play a major role in this case.     

A second order scheme for the navier-stokes equations: stability and convergence

In this paper, a fully discretized projection method is introduced. It contains a parameter operator. Depending on this operator, we can obtain a first-order scheme, which is appropriate for theoretical analysis, and a second-order scheme, which is more suitable for actual computations. In this method, the boundary conditions of the intermediate velocity field and pressure are not needed. We give the proof of the stability and convergence for the first-order case. For the higher order cases, the proof were different, and we will present it elsewhere.

In a forthcoming article [7], we apply this scheme to the driven-cavity problem and compare it with other schemes