Cartesian Mesh Research Articles

LES of a turbulent swirl flame using a mesh adaptive level-set method with dynamic load balancing

A turbulent lean premixed swirl flame is numerically investigated by solving the Navier–Stokes equations with a finite-volume large-eddy simulation (LES) method. A combined G-equation progress variable approach is applied to model the combustion process using a solution adaptive level-set solver. The finite-volume and the level-set solver are parallelized and coupled on a joint hierarchical Cartesian mesh, where each solver can individually use and adapt a subset of the mesh cells. The level-set solver adapts the mesh locally in a band close to the flame front location to automatically satisfy the high accuracy requirements of the level-set solution. The numerical results for the turbulent swirl flame are in excellent agreement with experimental findings. Since the flame shape varies during the computation, the number of cells associated with the flame shifts between the individual subdomains which changes the workload distribution on the parallel computing processes. To achieve a high parallel efficiency, a dynamic load balancing method is applied which determines a new partitioning of the grid using estimated cell weights based on measured computing times to redistribute the cells among all subdomains accordingly. The efficiency of the dynamic load balancing scheme to reduce load imbalances is demonstrated for a large-scale simulation of the investigated swirl flame on 170,000 compute cores. That is, the dynamic load balancing scheme reduces the computing time of this simulation by approximately 30%.

A geometrically accurate low-diffusive conservative interface capturing method suitable for multimaterial flows

This paper is dedicated to the design of an artifact-free and geometrically accurate low-diffusive interface capturing method for pure advection problems with extension to the multifluid/multimaterial case. The numerical method involves a genuinely multidimensional gradient limiting process that provides interface reconstruction accuracy without appearance of usually-encountered artifacts like zigzag-shaped instabilities or broken line-type attractors especially when Cartesian meshes are used. The approach is also extended to the multimaterial cases that gives very promising results. In particular, the well-known stretching Kothe-Rider test case has been extended up to five materials and provide convincing results.

Optimal Energy Conserving Local Discontinuous Galerkin Methods for Elastodynamics: Semi and Fully Discrete Error Analysis

We present an arbitrary high-order local discontinuous Galerkin (LDG) method with alternating fluxes for solving linear elastodynamics problems in isotropic media. Both the semi-discrete analysis and fully discrete analysis for a leap-frog LDG method are given to show that the proposed method simultaneously enjoys the energy conserving property and optimal convergence rates in both the displacement and stress, when the tensor product polynomials of the degree k are used on Cartesian meshes. Numerical experiments demonstrate that the proposed method has several advantages including the exact energy conservation, slow-growing errors in long time simulation, and subtle dependence on the first Lame parameter $$\lambda $$ .

A Two-Phase Numerical Model of Well Test Analysis to Characterize Formation Damage in Near-Well Regions of Injection Wells

Formation damage usually occurs in near-well regions for injection wells completed in offshore oilfields under the development of line drive patterns. However, current works on characterizing the damage by well test analysis were basically focused on using single-phase analogy to solve two-phase flow issues, resulting in errors on the diagnosis and interpretation of transient pressure data. In this paper, we developed a two-phase model to simulate the pressure transient behavior of a water injection well in a multiwell system. To solve the model more efficiently, we used the finite volume method to discretize partially differential flow equations in a hybrid grid system, including both Cartesian and radial meshes. The fully implicit Newton-Raphson method was also employed to solve the equations in our model. With this methodology, we compared the resulting solutions with a commercial simulator. Our results keep a good agreement with the solutions from the simulator. We then graphed the solutions on a log-log plot and concluded that the effects of transitional zone and interwell interference can be individually identified by analyzing specific flow regimes on the plot. Further, seven scenarios were raised to understand the parameters which dominate the pressure transient behavior of these flow regimes. Finally, we showed a workflow and verified the applicability of our model by demonstrating a case study in a Chinese offshore oilfield. Our model provides a useful tool to reduce errors in the interpretation of pressure transient data derived from injection wells located in a line drive pattern.

A Posteriori Subcell Finite Volume Limiter for General P_NP_M Schemes: Applications from Gasdynamics to Relativistic Magnetohydrodynamics

In this work, we consider the general family of the so called ADER P_NP_M schemes for the numerical solution of hyperbolic partial differential equations with arbitrary high order of accuracy in space and time. The family of one-step P_NP_M schemes was introduced in Dumbser (J Comput Phys 227:8209–8253, 2008) and represents a unified framework for classical high order Finite Volume (FV) schemes (N=0), the usual Discontinuous Galerkin (DG) methods (N=M), as well as a new class of intermediate hybrid schemes for which a reconstruction operator of degree M is applied over piecewise polynomial data of degree N with M>N. In all cases with M ge N > 0 the P_NP_M schemes are linear in the sense of Godunov (Math. USSR Sbornik 47:271–306, 1959), thus when considering phenomena characterized by discontinuities, spurious oscillations may appear and even destroy the simulation. Therefore, in this paper we present a new simple, robust and accurate a posteriori subcell finite volume limiting strategy that is valid for the entire class of P_NP_M schemes. The subcell FV limiter is activated only where it is needed, i.e. in the neighborhood of shocks or other discontinuities, and is able to maintain the resolution of the underlying high order P_NP_M schemes, due to the use of a rather fine subgrid of 2N+1 subcells per space dimension. The paper contains a wide set of test cases for different hyperbolic PDE systems, solved on adaptive Cartesian meshes that show the capabilities of the proposed method both on smooth and discontinuous problems, as well as the broad range of its applicability. The tests range from compressible gasdynamics over classical MHD to relativistic magnetohydrodynamics.

Оптимізація часу і статистичної похибки розрахунку біологічного захисту контейнерів зберігання відпрацьованого ядерного палива

The radiation protection is an important issue in the operation of nuclear power plants and artificial radioactive sources, which include spent nuclear fuel storage facilities. The Monte-Carlo codes are the effective instruments for calculation of radiation shielding properties and radiation field characteristics for complex geometries. However, achieving a satisfactory statistical error of the results in modeling the passage of neutrons and photons through biological protection may require excessively long calculation time. To solve this problem, Monte-Carlo codes use methods to reduce the variance to direct particles to regions with detectors to improve statistical accuracy. Our paper presents the application of the variance reduction function based on weight windows in the Monte-Carlo Serpent code, the function is investigated on the example of a simplified 2D model of the spent nuclear fuel storage container HI-STORM 190. The simple approach option of variance reduction with fixed cartesian and cylindrical meshes was investigated for different mesh nodes and for different dimensions of nodes. Also, global variance reduction option with fixed cartesian and cylindrical meshes was analyzed for case of achieving satisfactory results for the entire simulated volume. For a qualitative assessment of the variance reduction function, the indicator — figure of merit (FOM) used in our paper which proposed by the developers of the Serpent Monte-Carlo code. It is shown that the use of the variance reduction function leads to a significant decrease of statistical error and decrease of the calculation time, and therefore can be useful for biological protection calculations. As conclusions we can note that: the cylindrical mesh is not as effective in terms of FOM compared to Cartesian mesh; for both cylindrical and Cartesian meshes it is possible to find the recommended grain (node) size; the use of azimuthal partition of the cylindrical mesh together with radial partition leads to an increase in FOM; the application of global variance reduction is useful in the case of asymmetric biological protection geometries, while the FOM decreases.

Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh

Abstract The Poisson equation is commonly encountered in engineering, for instance, in computational fluid dynamics (CFD) where it is needed to compute corrections to the pressure field to ensure the incompressibility of the velocity field. In the present work, we propose a novel fully convolutional neural network (CNN) architecture to infer the solution of the Poisson equation on a 2D Cartesian grid with different resolutions given the right-hand side term, arbitrary boundary conditions, and grid parameters. It provides unprecedented versatility for a CNN approach dealing with partial differential equations. The boundary conditions are handled using a novel approach by decomposing the original Poisson problem into a homogeneous Poisson problem plus four inhomogeneous Laplace subproblems. The model is trained using a novel loss function approximating the continuous $ {L}^p $ norm between the prediction and the target. Even when predicting on grids denser than previously encountered, our model demonstrates encouraging capacity to reproduce the correct solution profile. The proposed model, which outperforms well-known neural network models, can be included in a CFD solver to help with solving the Poisson equation. Analytical test cases indicate that our CNN architecture is capable of predicting the correct solution of a Poisson problem with mean percentage errors below 10%, an improvement by comparison to the first step of conventional iterative methods. Predictions from our model, used as the initial guess to iterative algorithms like Multigrid, can reduce the root mean square error after a single iteration by more than 90% compared to a zero initial guess.

ADAPTIVE SOLUTION OF THE NEUTRON DIFFUSION EQUATION WITH HETEROGENEOUS COEFFICIENTS USING THE MIXED FINITE ELEMENT METHOD ON STRUCTURED MESHES

The neutron transport equation can be used to model the physics of the nuclear reactor core. Its solution depends on several variables and requires a lot of high precision computations. One can simplify this model to obtain the SPN equation for a generalized eigenvalue problem. In order to solve this eigenvalue problem, we usually use the inverse power iteration by solving a source problem at each iteration. Classically, this problem can be recast in a mixed variational form, and then discretized by using the Raviart-Thomas-Nédélec Finite Element. In this article, we focus on the steady-state diffusion equation with heterogeneous coefficients discretized on Cartesian meshes. In this situation, it is expected that the solution has low regularity. Therefore, it is necessary to refine at the singular regions to get better accuracy. The Adaptive Mesh Refinement (AMR) is one of the most effective ways to do that and to improve the computational time. The main ingredient for the refinement techniques is the use of a posteriori error estimates, which gives a rigorous upper bound of the error between the exact and numerical solution. This indicator allows to refine the mesh in the regions where the error is large. In this work, some mesh refinement strategies are proposed on the Cartesian mesh for the source problem. Moreover, we numerically investigate an algorithm which combines the AMR process with the inverse power iteration to handle the generalized eigenvalue problem.

APPLICATION OF THE SPH EQUIVALENCE TECHNIQUE TO THE CABRI FULL CORE IN NON-FUNDAMENTAL MODE

We developed a SPH equivalence technique in non-fundamental mode condition between a CABRI full-core model solved with the method of characteristics (MOC) in 2D and a simplified full-core model solved with the simplified P3 (SP3) method, linear anisotropic sources and discretized with Raviart-Thomas finite elements over a pure Cartesian mesh. The MOC and SP3 calculations are performed with DRAGON5 and DONJON5 codes, respectively. A three-parameter database is generated by DRAGON5 and is interpolated in DONJON5 as a function of the core condition. An objective function is set as the root mean square (RMS) error (MOC-SP3 discrepancy) on absorption distribution and leakage rates defined over the macro-geometry in DONJON5. Our algorithm is a quasi-Newtonian gradient search based on the Limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method. Numerical results are presented with Hafnium bars withdrawn or inserted.

Central discontinuous Galerkin methods on overlapping meshes for wave equations

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

A dynamic Immersed Boundary method for moving bodies and FSI applications

The document describes the CIRA and IBK collaborative research during the European Project UHURA. The latter aims to characterize the dynamic unsteady flow due to the deployment of a high-lift device at approach/landing conditions. The present work is devoted to the comprehension of the unsteady phenomena during the device motion by means of a coupled fluid-structure interaction (FSI) approach. In particular, the CIRA in-house immersed boundary (IB) code is extended to treat moving bodies and dynamic mesh refinements with the goal of simulating compressible and viscous flows around moving/deforming objects.The basic idea considers the motion of Lagrangian bodies through an inertial Cartesian mesh. Differently from classic moving-mesh approaches, the Cartesian cells do not move in space but rather they observe the solid walls crossing themselves. A moving least-square procedure (MLSQ) is used to build up a dynamic discrete forcing. The rigid body motion feature has been validated by simulating well-known benchmarks available in literature.A proper interface for coupling the IB method with a finite-element (FEM) solver is developed in the framework of a partitioned approach. In particular, the shared platform drives the solution sequence and allows exploring different coupling strategies. The aim is to simplify and automatize the study of complex flows whose characteristics strongly depends on the structural properties of moving objects. The final application deals with the FSI simulation of a high-lift device during its deployment phase.

ITERATIVE AND PARALLEL PERFORMANCE ANALYSIS OF NON-BLOCKING COMMUNICATION ALGORITHMS IN THE MASSIVELY PARALLEL NEUTRON TRANSPORT CODE PIDOTS

The PIDOTS neutral particle transport code utilizes a red/black implementation of the Parallel Gauss-Seidel algorithm to solve the SN approximation of the neutron transport equation on 3D Cartesian meshes. PIDOTS is designed for execution on massively parallel platforms and is capable of using the full resources of modern, leadership class high performance computers. Initial testing revealed that some configurations of PIDOTS’s Integral Transport Matrix Method solver demonstrated unexpectedly poor parallel scaling. Work at Idaho and Los Alamos National Laboratories then revealed that this inefficiency was a result of the accumulation of high-cost latency events in the complex blocking communication networks employed during each PIDOTS iteration. That work explored the possibility of minimizing those inefficiencies while maintaining a blocking communications model. While significant speedups were obtained, it was shown that fully mitigating the problem on general-purpose platforms was highly unlikely for a blocking code. This work continues that analysis by implementing a deeply interleaved non-blocking communication model into PIDOTS. This new model benefits from the optimization work performed on the blocking model while also providing significant opportunities to overlap the remaining un-mitigated communication costs with computation. Additionally, our new approach is easily transferable to other similarly spatially decomposed codes. The resulting algorithm was tested on LANL’s Trinity system at up to 32,768 processors and was found at that processor count to effectively hide 100% of MPI communication cost – equivalently 20% of the red/black phase time. It is expected that the implemented interleaving algorithm can fully support far higher processor counts and completely hide communication costs up ~50% of total iteration time.

Large-Eddy Simulations of Rim Seal Flow in a One-Stage Axial Turbine

The flow field in a one-stage axial flow turbine with 30 stator and 62 rotor blades including the wheel space is investigated by large-eddy simulation (LES). The Navier-Stokes equations are solved using a massively parallel finite-volume solver based on a Cartesian mesh with immersed boundaries. The strict conservation of mass, momentum, and energy is ensured by an efficient cut-cell/level-set ansatz, where a separate level-set solver describes the motion of the rotor. Both solvers use individual subsets of a shared Cartesian mesh, which they can adapt independently. The focus of the analysis is on the flow field inside the rotor stator cavity between the stator and rotor disks. Two cooling gas mass flow rates are investigated for the same rim seal geometry. First, the time averaged flow field for both simulations is compared, followed by a detailed investigation of the unsteady flow field. The results for the cooling effectiveness are compared to experimental data. Both cases show good agreement with experimental data. It is shown that for the lower cooling gas mass flux several of the wheel space’s acoustic waves are excited. This is not observed for the higher cooling gas mass flux. The excited waves lead to stable, i.e., bounded, fluctuations inside the wheel space and result in a significantly higher hot gas ingestion.

New immersed finite volume element method for elliptic interface problems with non-homogeneous jump conditions

In this paper, we propose a new immersed finite volume element method to solve elliptic problems with discontinuous diffusion coefficient and sharp-edged interfaces on Cartesian mesh. The method uses the non-traditional immersed finite volume element method together with additional immersed finite element function on interface element. It can deal with the case when the solution or its normal derivative is discontinuous. Extensive numerical experiments for various problems show that the new method is approximate second-order convergence in the L∞ norm for piecewise smooth solutions, and more than 1.65th order accuracy is observed for solution with singularity.

Calculation of the Plastic Metal Flow in the Cold Extrusion Technology by Finite Volume Method

This paper presents an application of the finite volume method to ideal plastic metal flow in extrusion technology. Governing equations for the mass and momentum balance are used in an integral form. Solution domains in the cases considered are discretized with a Cartesian numerical mesh with computational points placed at the center of each control volume. After discretization of the governing equations, the resulting system of nonlinear algebraic equations is solved by an iterative procedure, using a segregated algorithm approach. Resulting stress fields are obtained from the Levy-Mises equations. The experimental results and numerical calculations are in good agreement.

An efficient second order all Mach finite volume solver for the compressible Navier–Stokes equations

In the numerical simulation of fluid dynamic problems there are situations in which acoustic waves are very fast compared to the average velocity of the fluid and conversely situations in which the fluid moves at high speed and shock waves may be present. Ideally, a numerical method should be able to treat these different regimes without strong limitations in terms of time step and without excessive related computational cost. Unfortunately, standard explicit in time schemes often adopted for hyperbolic problems are not suitable for these problems, hence remedies have to be studied. To this aim, the results presented in this article concern the development of a second order in time and space numerical method for the compressible Navier–Stokes equation which works for both high and low Mach numbers. In particular, when the Mach number goes to zero, one recovers a numerical method for the limit Navier–Stokes system which under some additional hypothesis degenerates to the incompressible Navier–Stokes equations, while in the case of high Mach numbers the method exhibits a shock capturing structure. The idea is based on partitioning the equations into a fast and a slow scale and by taking implicit the fast scale dynamic together with the viscous terms. The resulting numerical scheme is stable for time steps which are independent both from the speed of the pressure waves and from the diffusive terms characterizing the viscous forces and the heat flux. The only time step limitation is induced by the average speed of the flow. The work here presented extends the seminal ideas developed in Dimarco et al. (2017, 2018) for isentropic Euler equations and in Boscheri et al. (2020) for the full set of compressible Euler equations to the multidimensional Navier–Stokes system and permits efficient three dimensional simulations of all Mach problems. The discretization is constructed on Cartesian meshes and the method is second order accurate in space and time. Numerical results show the accuracy, the robustness and the effectiveness of the new proposed approach.

Porous plasticity revisited: Macroscopic and multiscale modeling

Porous plasticity aims to model the growth and coalescence of voids leading to ductile failure. The GTN model (1984), resulting from heuristic modifications to Gurson's homogenized hollow sphere model (1977), is used in numerous publications. The Rousselier model (1981), developed in the framework of continuum thermodynamics, is apparently similar. Both models are effective in numerical calculations, but the reasons why they perform well were not investigated in details in the existing literature, as regards transition to uniaxial deformation, relations between various modes of strain localization, finite element discretization, regularization. In the present paper, we propose first to revisit both models and to compare their fundamentally different mechanical behaviors. For stress triaxiality larger than some critical value, it is shown that theoretically the GTN model cannot achieve strain localization in a plane but only pointwise localization for the ultimate mechanical state (stress tensor equal to zero). The larger the void volume fraction (void growth), the smaller the stress triaxiality critical value. Fortunately, discretization transforms the pointwise localization into volume localization and with an appropriate Cartesian finite element mesh a more or less planar sheet of integration points can be obtained. The Rousselier model can achieve strain localization in a plane at all stress triaxialities and discretization also transforms this localization into volume localization with a characteristic element size. Second, multiscale modeling of both plasticity and ductile damage (not limited to void damage) is an essential way of progress for laboratory specimen calculations. The Rousselier model can be incorporated into polycrystalline models based on crystal plasticity, with reasonable computation times provided a reduced texture with a small number of crystallographic orientations is used. It can be coupled with a new Coulomb ductile fracture model at the slip system scale and with secondary void nucleation and growth models at the grain and slip system scales, respectively. The multiscale model is applied to aluminum CT and KAHN specimens and to steel round notched specimens.

An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f(u) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f′(u) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32]. The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

On the design of potential turbine positions for physics-informed optimization of wind farm layout

Wind farm layout optimization is a critical step in the design of a wind energy project. In the literature, the potential turbine positions employed in the layout optimization are often obtained by discretizing the field using a Cartesian mesh. In this work, physical understanding is proposed to incorporate in the design of potential turbine positions. Specifically, the known knowledge, that a staggered arrangement is more efficient for extracting energy from wind than an aligned arrangement, is employed and implemented using the staggered mesh approach, the unstructured mesh approach and the sunflower mesh approach. Different mesh approaches are tested using two cases, i.e. case I, unidirectional uniform wind, and case II, uniform wind with variable wind direction. The optimal layout obtained from the staggered mesh approach performs the best for case I. For case II, the farm performance from different layouts is similar. The performance of the layouts under off-design conditions is also tested for case I. For all considered cases, the optimal layout obtained from the sunflower approach shows an overall good performance.

Scale-adaptive turbulence modeling for LES over complex terrain

The large-eddy simulation (LES) is an efficient method for the study of atmospheric boundary layer (ABL) flow over complex terrain. However, the robustness of LES is influenced by dynamics methods for the subgrid-scale stress, which remains one of the challenging issues. Here, we present a scale-adaptive LES methodology to study ABL flow over complex terrain. We present a canopy stress method for LES over complex terrain, which provides no-slip and shear-stress boundary conditions using a cost-effective Cartesian mesh. We consider measurements from wind tunnel experiments for the investigation of a scale-adaptive subgrid-scale modeling framework for LES of wind flow over a hilly terrain. A flow simulation is then performed on six cases of ABL flow over an idealized complex terrain. The LES result is also compared against the measurement of wind over Askervein hill. The dynamic and non-dynamic versions of the turbulence kinetic energy-based subgrid model are also considered to illustrate the scale-adaptive nature of subgrid-scale turbulence. It was observed that ignoring scale-adaptivity of subgrid turbulence, LES can under-predict shear-stress, turbulence kinetic energy, and other statistical measures of atmospheric turbulence.