Mesh Velocity Research Articles

An Adaptive Moving Mesh Method for Forced Curve Shortening Flow.

We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretize in space the governing evolution equation for the position vector, and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes.

Continuum shape sensitivity analysis for aeroelastic gust using an arbitrary lagrangian-eulerian reference frame

Continuum Sensitivity Analysis (CSA), a method to determine response derivatives with respect to design variables, is derived here for the first time in an arbitrary Lagrangian-Eulerian (ALE) reference frame. CSA differentiates nonlinear governing system of equations to arrive at a linear system of partial differential continuum sensitivity equations (CSEs), here, for fluid-structure interaction (FSI). The CSEs and associated sensitivity boundary conditions are derived here for the first time for FSI, using the boundary velocity formulation, carefully distinguishing design velocity from flow velocity and ALE mesh velocity. Whereas boundary conditions must be differentiated using the material (total) derivative, it is sometimes advantageous to derive the CSEs using local (partial) derivatives. The benefit is that geometric sensitivity, known as design velocity, may not be required in the domain. It is shown here that this advantage is realized when the ALE frame undergoes only the rigid body motion associated with the structure to which it is attached. It is further shown that the advantage is not realized when the ALE mesh deforms due to the flexible motion of the fluid-structure interface. The equations for the transient gust response of a two-dimensional airfoil in compressible flow, flexibly attached to a rigid body mass, are presented as a model problem to illustrate a detailed derivation.

Interface‐fitted moving mesh method for axisymmetric two‐phase flow in microchannels

SummaryA boundary‐fitted moving mesh scheme is presented for the simulation of two‐phase flow in two‐dimensional and axisymmetric geometries. The incompressible Navier‐Stokes equations are solved using the finite element method, and the mini element is used to satisfy the inf‐sup condition. The interface between the phases is represented explicitly by an interface adapted mesh, thus allowing a sharp transition of the fluid properties. Surface tension is modelled as a volume force and is discretized in a consistent manner, thus allowing to obtain exact equilibrium (up to rounding errors) with the pressure gradient. This is demonstrated for a spherical droplet moving in a constant flow field. The curvature of the interface, required for the surface tension term, is efficiently computed with simple but very accurate geometric formulas. An adaptive moving mesh technique, where smoothing mesh velocities and remeshing are used to preserve the mesh quality, is developed and presented. Mesh refinement strategies, allowing tailoring of the refinement of the computational mesh, are also discussed. Accuracy and robustness of the present method are demonstrated on several validation test cases. The method is developed with the prospect of being applied to microfluidic flows and the simulation of microchannel evaporators used for electronics cooling. Therefore, the simulation results for the flow of a bubble in a microchannel are presented and compared to experimental data.

Three-Dimensional Aeroelastic Solutions via the Nonlinear Frequency-Domain Method

An aeroelastic solver is developed using a nonlinear frequency domain flow solver coupled to a plate-bending finite element linear structural solver. A methodology for determining the flow conditions leading to flutter and limit-cycle oscillations is proposed, based on a root-finding Newton–Raphson iterative method. The novelty of the approach lies in the constant size of the Newton–Raphson system of equations, regardless of the number of degrees of freedom of the structural model. To serve this purpose, a new method of computing mesh velocities for nonlinear frequency-domain flow solvers is developed, and a technique for solving the geometric conservation law within the nonlinear frequency-domain framework is presented accordingly. The new approach for computing mesh velocities is validated against existing experimental data on the Lockheed, Air Force, NASA and Netherlands wing (run 73, CT5), whereas the aeroelastic solver is validated via experimental results of the AGARD I.-wing weakened model 3 and solid model 2 in air and R-12, respectively. The proposed framework is expected to perform limit-cycle oscillation computations an order of magnitude faster than a typical aeroelastic time-marching approach.

Improved ALE mesh velocities for complex flows

SummaryA key choice in the development of arbitrary Lagrangian‐Eulerian solution algorithms is how to move the computational mesh. The most common approaches are smoothing and relaxation techniques, or to compute a mesh velocity field that produces smooth mesh displacements. We present a method in which the mesh velocity is specified by the irrotational component of the fluid velocity as computed from a Helmholtz decomposition, and excess compression of mesh cells is treated through a noniterative, local spring‐force model. This approach allows distinct and separate control over rotational and translational modes. The utility of the new mesh motion algorithm is demonstrated on a number of 3D test problems, including problems that involve both shocks and significant amounts of vorticity.

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

SummaryThe Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.

ALE-SUPG finite element method for convection–diffusion problems in time-dependent domains: Conservative form

A Streamline Upwind Petrov–Galerkin (SUPG) finite element method for a convection dominated transient convection-diffusion-reaction equation in time-dependent domains is proposed. The time-dependent domain is handled by the arbitrary Lagrangian–Eulerian (ALE) approach, whereas the SUPG method is used for the spatial discretization. Further, the first order modified backward Euler and the second order modified Crank–Nicolson methods are used for the temporal discretization. It is shown that the stability of the semi-discrete (continuous in time) conservative ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete scheme with the implicit Euler time discretization is unconditionally stable and is only conditionally stable (time step depends on mesh velocity) with the Crank–Nicolson method. Numerical results are presented to support the stability estimates and to show the influence of the SUPG stabilization parameter in a time-dependent domain. Further, the proposed numerical scheme is applied to a boundary/layer problem in a time-dependent domain.

A study on moving mesh finite element solution of the porous medium equation

An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two-dimensional numerical results are presented.

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

SummaryIn this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step.We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.

High Order Direct Arbitrary-Lagrangian–Eulerian (ALE) Finite Volume Schemes for Hyperbolic Systems on Unstructured Meshes

In this work we develop a new class of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space–time finite element predictor, which performs a one-step time discretization. Specifically, we adopt a discontinuous Galerkin predictor which can handle stiff source terms that might produce jumps in the local space–time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space–time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space–time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space–time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The nodal solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration at time $$t^n$$ is connected with the new one at time $$t^{n+1}$$ with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space–time divergence operator and is then integrated on a moving space–time control volume. We adopt a linear parametrization of the space–time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical magneto-hydrodynamics equations and the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Finally we briefly present some variants of the algorithm that aim at improving the overall computational efficiency.

Arbitrary Lagrangian Eulerian Formulation for Sloshing Tank Analysis in Nuclear Engineering

Fluid-structure interaction plays an important role in nuclear engineering design, where several numerical and experimental tests need to be performed on new tank design before getting into the production process. The design can be performed for fluid storage tanks that require knowledge of sloshing frequencies and hydrodynamic pressure distribution on the structure. These can be very useful for engineers and designers to define appropriate material properties and shell thickness of the structure to be resistant under seismic loading. Data presented in current tank seismic design codes such as Eurocode are based on simplified assumptions for the geometry and material tank properties. Fuel tanks may undergo different types of loading, including seismic loading, where the behavior of storage tanks includes material nonlinearities, which are caused by material yielding. The Arbitrary Lagrangian Eulerian formulation based on finite element analysis presented in the paper takes into account material properties of the structure as well as the complex geometry of the tank. The formulation uses a moving mesh with a mesh velocity defined through the structure motion. In this paper, we use different approaches to solve a fluid-structure coupling problem. The first one uses the full Navier-Stokes equation for the fluid with projection method, and the second approach uses potential flow theory. The problem consists of a sloshing deformable tank submitted to acceleration loading.

Tidal current energy potential assessment by a two dimensional computational fluid dynamics model: The case of Avilés port (Spain)

Tidal energy has significant potential yet to be developed. One of the key areas of the research lines being developed is the ability to perform accurate computational models in order to make a full appraisal of kinetic energy potential in different areas through tide movements. So far, due to the limitations of computational resources and the extension of the geographical domains, existing models have manly used simplifications such as low resolution mesh sizes and velocity profile approximations in the vertical dimension. The investigation put forward includes a high resolution two dimensional equivalent model of the zone studied using a longitudinal section with depth and length dimensions, as well as a simulation methodology to study the velocity field in different sections. The model can be used in cases of channels of high width compared to the depth. With such data, kinetic power and energy of tidal currents can be predicted. Both, model and methodology have been applied and validated to the specific case of the estuary of Avilés port, thereby obtaining a new perspective in the hydrodynamic study as well as a tidal current energy potential assessment.

A multi-dimensional high-order DG-ALE method based on gas-kinetic theory with application to oscillating bodies

This paper presents a multi-dimensional high-order discontinuous Galerkin (DG) method in an arbitrary Lagrangian-Eulerian (ALE) formulation to simulate flows over variable domains with moving and deforming meshes. It is an extension of the gas-kinetic DG method proposed by the authors for static domains (X. Ren et al., 2015 22). A moving mesh gas kinetic DG method is proposed for both inviscid and viscous flow computations. A flux integration method across a translating and deforming cell interface has been constructed. Differently from the previous ALE-type gas kinetic method with piecewise constant mesh velocity at each cell interface within each time step, the mesh velocity variation inside a cell and the mesh moving and rotating at a cell interface have been accounted for in the finite element framework. As a result, the current scheme is applicable for any kind of mesh movement, such as translation, rotation, and deformation. The accuracy and robustness of the scheme have been improved significantly in the oscillating airfoil calculations. All computations are conducted in a physical domain rather than in a reference domain, and the basis functions move with the grid movement. Therefore, the numerical scheme can preserve the uniform flow automatically, and satisfy the geometric conservation law (GCL). The numerical accuracy can be maintained even for a largely moving and deforming mesh. Several test cases are presented to demonstrate the performance of the gas-kinetic DG-ALE method.

SPH and ALE formulations for sloshing tank analysis

Design of fuel tanks requires the knowledge of hydrodynamic pressure distribution on the structure. These can be very useful for engineers and designers to define appropriate material properties and shell thickness of the structure to be resistant under sloshing or hydrodynamic loading. Data presented in current tank design codes as Eurocode, are based on simplified assumptions for the geometry and material tank properties. For complex material data and complex tank geometry, numerical simulations need to performed in order to reduce experimental tests that are costly and take longer time to setup. Different formulations have been used for sloshing tank analysis, including ALE (Arbitrary Lagrangian Eulerian) and SPH (Smooth Particle Hydrodynamic). The ALE formulation uses a moving mesh with a mesh velocity defined trough the structure motion. In this paper the mathematical and numerical implementation of the FEM and SPH formulations for sloshing problem are described. From different simulations, it has been observed that for the SPH method to provide similar results as ALE formulation, the SPH meshing, or SPH particle spacing needs to be finer than ALE mesh. To validate the statement, we perform a simulation of a sloshing analysis inside a partially filled tank. For this simple, the particle spacing of SPH method needs to be at least two times finer than ALE mesh. A contact algorithm is performed at the fluid structure interface and SPH particles. In the paper the efficiency and usefulness of two methods, often used in numerical simulations, are compared.

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved at the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge–Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.

Generalized coordinate transformation and gas-kinetic scheme

Under a generalized coordinate transformation with inclusion of both mesh velocity and acceleration, the gas-kinetic equation is transformed from a physical space to a computational space. With the WENO reconstruction, a high-order moving-mesh multidimensional gas-kinetic method, with inclusion of both normal and tangential derivatives of macroscopic flow variables, is constructed. High-order accurate Navier–Stokes solutions are obtained in the current scheme by integrating the numerical fluxes along a cell interface and over a whole time step, without using the conventional Gaussian point integration for spatial accuracy and Runge–Kutta method for temporal accuracy. For the first time, the mesh acceleration has been explicitly accounted for in the flux evaluation. Under the generalized coordinate transformation, a three-dimensional moving-mesh gas-kinetic scheme has been constructed as well. The importance of multi-dimensionality in the flux evaluation and mesh movement is illustrated through numerical tests.

High‐order time integrators for front‐tracking finite‐element analysis of viscous free‐surface flows

SummaryThis paper proposes implicit Runge–Kutta (IRK) time integrators to improve the accuracy of a front‐tracking finite‐element method for viscous free‐surface flow predictions. In the front‐tracking approach, the modeling equations must be solved on a moving domain, which is usually performed using an arbitrary Lagrangian–Eulerian (ALE) frame of reference. One of the main difficulties associated with the ALE formulation is related to the accuracy of the time integration procedure. Indeed, most formulations reported in the literature are limited to second‐order accurate time integrators at best. In this paper, we present a finite‐element ALE formulation in which a consistent evaluation of the mesh velocity and its divergence guarantees satisfaction of the discrete geometrical conservation law. More importantly, it also ensures that the high‐order fixed mesh temporal accuracy of time integrators is preserved on deforming grids. It is combined with the use of a family of L‐stable IRK time integrators for the incompressible Navier–Stokes equations to yield high‐order time‐accurate free‐surface simulations. This is demonstrated in the paper using the method of manufactured solution in space and time as recommended in Verification and Validation. In particular, we report up to fifth‐order accuracy in time. The proposed free‐surface front‐tracking approach is then validated against cases of practical interest such as sloshing in a tank, solitary waves propagation, and coupled interaction between a wave and a submerged cylinder. Copyright © 2015 John Wiley & Sons, Ltd.

Adjoint shape design sensitivity analysis of fluid–solid interactions using concurrent mesh velocity in ALE formulation

A coupled variational equation for fluid–solid interaction (FSI) problems is derived using a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, a traction continuity condition at interfaces, and a pseudo-equilibrium equation for mesh velocity. The moving boundary in arbitrary Lagrangian–Eulerian (ALE) formulation is included in the variational equations by the mesh velocity obtained from a displacement-loaded pseudo-structural problem at a concurrent configuration, which eventually facilitates to derive shape design sensitivity. A continuum-based adjoint shape sensitivity is derived under ALE formulation, which turns out to be very accurate and efficient due to the utilization of converged tangent and the linearity of both adjoint and sensitivity equations. Through numerical examples, the obtained sensitivity is verified in terms of accuracy and efficiency compared with finite difference sensitivity and further applied to the shape optimization problem of finding a stiff structure while satisfying a volume constraint.

A three-dimensional finite element arbitrary Lagrangian–Eulerian method for shock hydrodynamics on unstructured grids

We present a three-dimensional (3D) finite element (FE) arbitrary Lagrangian–Eulerian (ALE) method for shock hydrodynamics on unstructured grids. The method is based on an FE Eulerian Godunov scheme for linear tetrahedra that has been extended to include mesh motion in an unsplit, flux-conservative formulation. The proposed method eliminates the splitting errors present in traditional Lagrange-plus-remap methods that occur during the remap phase. Unlike typical unsplit approaches, the mesh velocity is not determined by boundary motion but is instead based on the local fluid velocity. Smoothing operations are then applied to the mesh velocity to avoid mesh tangling. This approach allows the mesh to follow the fluid motion in a robust manner and leverage one of the primary advantages of Lagrangian schemes for shock hydrodynamics, namely that the resolution follows the flow. An approximate Riemann solver is used to calculate fluxes in the co-moving frame of the mesh. Results for a number of standard test problems are presented for 3D meshes of up to 107 tetrahedra. Global convergence rates of 0.8–1.0 are observed for shock dominated flows and 1.9 for smooth flows. We also demonstrate that the method satisfies the discrete geometric conservation law to truncation error, conserves total energy to machine precision, and preserves symmetry.

Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

AbstractIn this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor proposed in. For that purpose, a new element-local weak formulation of the governing PDE is adopted on moving space-time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes withstraightedges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.