Computational Solver Research Articles

Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier–Stokes equations. Our method employs a technique developed by Guermond and Minev (2011), which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step, we solve a PDE having weak-derivatives in one direction only (which allows for the linear computational cost), at the expense of handling strong second-order derivatives of the previous time step solution, on the right-hand side of these PDEs. This motivates the use of smooth functions such as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To deal robustly with these instabilities, we propose to use a residual minimization technique. We test our method on problems having manufactured solutions, as well as on the cavity flow problem.

Unsteady 2D and 3D Navier-Stokes Solver with Application of Multigrid Scheme to Pressure Poisson Fractional Step on Arbitrary Unstructured Grids in Various Applications with Emphasis on Ship Motion

A 3D unsteady computer solver is presented to compute incompressible Navier-Stokes equations combined with the volume of fraction (VOF) method on an arbitrary unstructured domain. This is done to simulate fluid flows in various applications, especially around a marine vessel. The Navier-Stokes solver is based on the fractional steps method coupled with a finite volume scheme and collocated grids by which velocity components and pressure fields are defined at the center of the control volume. However, the fluxes are defined at the midpoint on their corresponding cell faces. On the other hand, the CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) scheme is applied to capture the free surface. In the presented fractional step method, the pressure Poisson equation suffers from poor convergence rate by simple iterative methods like Successive Overrelaxation (SOR), especially in simulating complex geometrics like a ship with appendages. Therefore, to accelerate the convergence rate, an agglomeration multigrid method is applied on arbitrary moving mesh for solving pressure Poisson equation with two well-known cycles, V and W. In order to maintain accuracy, the geometry details should not change in grid coarsening procedure. Therefore, the boundary faces are assumed to be fixed in all grids level. This assumption requires nonstandard cells in coarsening procedures. To investigate the performance of the applied algorithm, various flows including one and two-phase flows are studied in two and three dimensions. It is found that the multigrid method can speed up the convergence rate of fractional step twofold. In most cases (not all), W cycle displays better performance. It is also concluded that the efficiency of the cycle depends on the number of meshes and complexity of the problem and this is mainly due to the data transferring between grids. Therefore, the type of cycle should be selected judiciously and carefully, while considering the mesh size and flow properties.

Treating linear molecules in calculations of rotation-vibration spectra.

In this article, a numerical implementation of the exact kinetic energy operator (KEO) for triatomic molecules (symmetric of XY2-type and asymmetric of YXZ-type) is presented. The implementation is based on the valence coordinates with the bisecting (XY2-type molecules) and bond-vector (YXZ) embeddings and includes the treatment of the singularity at linear geometry. The KEO is represented in a sum-of-product form. The singularity caused by the undetermined angle at the linear configuration is resolved with the help of the associated Legendre and Laguerre polynomials used as parameterized bending basis functions in the finite basis set representation. The exact KEO implementation is combined with the variational solver theoretical rovibrational energies, equipped with a general automatic symmetry-adaptation procedure and efficient basis step contraction schemes, providing a powerful computational solver of triatomic molecules for accurate computations of highly excited ro-vibrational spectra. The advantages of different basis set choices are discussed. Examples of specific applications for computing hot spectra of linear molecules are given.

An Adaptively-damped Compressible-liquid Model for Non-cavitating Hydraulic Surges

This research presents a compact and computationally-efficient two-equation compressible-liquid model. The model is specifically developed for the numerical computation of hydraulic surges in pipes under high fluid pressure where cavitation is absent. The proposed model aims to simplify the three-equation model of Neuhaus et al. for two-phase cavitational hammers. Compressible effects in liquid during the transients are considered by including a suitable equation of state into the model. A tunable function of the relative local pressure fluctuation called 'Variable Friction Coefficient' (VFC) for the flow transients is also incorporated into the model. For the accurate modeling of wave propagation, the split-coefficient matrix (SCM) method for characteristic-direction based splitting of eigenvalues is used in the study. The results show that the proposed two-equation model can reproduce the results from the three-equation model at a substantially reduced computational cost. The integration of the variable friction coefficient into the two-equation compressible-liquid model further improved the solver capability. The results computed using this aggregate solver are superior to the original three-equation model and the two-equation model without VFC. The results also suggest that the variable friction coefficient imparts adaptive damping capability to the solver model. This feature of the model is visible in the improved accuracy in the modeling of decaying pressure waves. The aggregate solver model, i.e., `the variable friction coefficient integrated two-equation compressible-liquid model,' offers a greatly simplified mathematical model and an inexpensive computational solver for the simulation of hydraulic surges in non -cavitating flow transients.

Linear computational cost implicit solver for elliptic problems

In this paper, we use the alternating direction method for isogeometric finite elements to simulate implicit dynamics. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit marching method to fully discretize the problem. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along a particular coordinate axis in the intermediate time steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of this algebraic transformations, we get a system of linear equations that can be factorized in linear $O(N)$ computational cost in every time step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

Oxygen transport in a permeable model of abdominal aortic aneurysm

This paper reports on modelling and simulation of the coupled mass and momentum transport through the arterial lumen and the porous arterial wall of an abdominal aortic aneurysm (AAA). The effect of porous structure and intramural flow, which is usually neglected, is essential to quantify the oxygen concentration in the aneurysmal wall and will be addressed in this work via a novel coupled fluid–porous structure–mass transport model for AAA. The computational solver provides a prediction of the filtration velocity and oxygen concentration in the artery and further simulations are used to investigate the relation between the wall oxygen concentration and permeability. The results demonstrate the essential coupling between blood and filtration flow and specifically how their interactions affect oxygen transport.

A three-dimensional unified gas-kinetic wave-particle solver for flow computation in all regimes

In this paper, the unified gas-kinetic wave-particle (UGKWP) method has been constructed on a three-dimensional unstructured mesh with parallel computing for multiscale flow simulation. Based on the direct modeling methodology, the unified gas-kinetic scheme (UGKS) models the flow dynamics directly on the numerical mesh size and time step scales, and it is able to capture the flow dynamics from the kinetic scale particle transport to the hydrodynamic wave propagation seamlessly according to the local cell Knudsen number. Instead of discretizing the particle velocity space in UGKS, the UGKWP method is composed of evolution of deterministic wave and stochastic particles. With dynamic wave-particle decomposition according to the cell Knudsen number, the UGKWP method is able to capture the continuum wave interaction and rarefied particle transport under a unified framework and achieves high efficiency in different flow regimes. The UGKWP flow solver is constructed in three-dimensional space and is validated by many test cases at different Mach and Knudsen numbers. The examples include a 3D shock tube problem, lid-driven cubic cavity flow, high-speed flow passing through a cubic object, and hypersonic flow around a space vehicle. The parallel performance has been tested on the Tianhe-2 supercomputer, and reasonable parallel performance has been observed up to 1000 cores. With the wave-particle formulation, the UGKWP method has great potential in solving three-dimensional multiscale transport problems with the co-existence of continuum and rarefied flow regimes, especially for the high-speed rarefied and continuum flow simulation around a space vehicle in near-space flight, where the local Knudsen number can vary significantly with five or six orders of magnitude differences.

A stochastic computational intelligent solver for numerical treatment of mosquito dispersal model in a heterogeneous environment

In this paper, the design of stochastic computational intelligent solver is presented for the solution of mathematical model representing the dynamics of mosquito dispersal in a heterogeneous environment by feedforward artificial neural networks (FFANNs) trained with genetic algorithms (GAs) aided with sequential quadratic programming (SQP), i.e., FFANN-GASQP. In the scheme FFANN-GASQP, the formulation of fitness function in mean square error sense with continuous mapping-based differential equation models of FFANNs for the mosquito dispersal system and training of these networks are accomplished by integrated competency of GA and SQP. The exactness, reliability and stability of the designed FFANN-GASQP approach are established through comparative studies with Adams numerical results for both single and multiple runs. Outcomes of statistical assessments are used to validate the accuracy and convergence of the designed FFANN-GASQP scheme.

Speed control of permanent magnet synchronous motor using neural network model predictive control

Model predictive control has been widely used in the industry. This can control the multivariable system with constraints on input and output variables but it needs online computation solver, and creates the non-convex solution in nonlinear plant due to the parameter uncertainties. The online computational problem and non-convex solution of the model predictive control are achieved via neural network model predictive control. The paper explores the speed control of permanent magnet synchronous motor (PMSM) by using neural network model predictive control (NNMPC) technique. The multi-layer artificial neural network is used to identify the dynamics of PMSM. The set point speed tracking control of PMSM is identified by using neural network model predictive control strategy. By using the set of input and output data obtained from the system, the multi input-output feed-forward neural network model is created. Levenberg-Marquardt algorithm is used to train the process models of the PMSM. That provides future plant output for control optimization of the predictive control. The overall system is developed and tested in the MATLAB/Simulink. To evaluate the efficiency of the controller proposed, it is compared with a constrained model predictive controller through the studies of simulation. The overshoot and settling time of the speed response of the PMSM are measured and analyzed for NNMPC and constrained MPC.

Flow transitions and mapping for undulating swimmers

Natural swimmers usually perform undulations to propel themselves and perform a range of maneuvers. These include various biological species ranging from micro-sized organisms to large-sized fishes that undulate at typical kinematic patterns. In this paper, we consider anguilliform and carangiform swimming modes to perform numerical simulations using an immersed-boundary methods based computational solver at various Reynolds number regimes. We carry out thorough studies using wavelength and Strouhal frequency as the governing parameters for the hydrodynamic performance of undulating swimmers. Our analysis shows that the anguilliform kinematics achieves better hydrodynamic efficiency for viscous flow regime, whereas for flows with higher Reynolds number, the wavelength of the wavy motion dictates which kinematics will outperform the other. We find that the constructive interference between vortices produced at anterior parts of the bodies and co-rotating vortices present at the posterior parts plays an important role in reversing the direction of Benard-von Karman vortex street. Since most of the thrust producing conditions appear to cause wake deflection; a critical factor responsible for degrading the hydrodynamic efficiency of a swimmer, we discuss the underlying mechanics that would trigger this phenomenon. We demonstrate that the choice of kinematic and flow conditions may be restricted for the natural swimmers due to their morphological structures, but our findings provide a guideline on their selection for bio-inspired underwater vehicles.

Numerical investigation of the effects of fan-coil airflow direction on distribution and deposition of indoor pollutant particles

Given the high concentration of particulate pollutants in the indoor environment compared to outdoor environment and their harmful effects on human health, it is important to examine the effect of indoor ventilation systems on distribution and deposition of particulate pollutants. Floor-mounted fan-coils have various airflow directions, but the effects of different directions on particle distribution and deposition have not been investigated in previous studies. Thus, the present study aims to investigate the indoor air quality in a room with vertical, 60° inclined, 30° inclined, and horizontal discharge fan-coils. Accordingly, the concentration of particles with a uniform initial distribution was evaluated in the room while considering the occupants’ thermal comfort conditions for four different modes of airflow using computational fluid dynamics and solver packages developed in OpenFOAM® solver by the authors. The particles with 2.5, 10, and 25 μm diameters were considered to examine the effect of particles size on their distribution and deposition. The results indicated that the percentages of 2.5 μm filtered particles for vertical, 60° inclined, 30° inclined, and horizontal fan-coil discharge modes after 600 s were about 93, 84, 76 and 93%, respectively. Also, for all discharge modes, the percentage of 2.5 μm deposited particles on the floor was less than 1%. The results also indicated that after about 975, 1800, 1390 and 940 s, the concentration of 2.5 μm particles reached 1% of their initial concentration, for vertical, 60° inclined, 30° inclined and horizontal fan-coil discharge modes respectively. As the particle size increased, the percentage of filtered particles decreased, while the percentage of particles deposited on the floor increased. Specifically, the percentages of deposited 25 μm particles under the vertical, 60° inclined, 30° inclined and horizontal discharge modes were about 36, 45, 66, and 49%, respectively.

NUMERICAL ANALYSIS OF THE IMPACT OF SIDESLIP ANGLE ON LOAD OF THE GYROCOPTER STABILIZERS

The paper presents some of the works related to the project of modern gyrocopter construction with the possibility of a short start, known as "jump-start". It also presents a methodology related to numerical calculations using Computational Fluid Dynamics based on ANSYS Fluent three-dimensional solver. The purpose of the work was to calculate the forces and aerodynamic moments acting on the gyrocopter stabilizers. The calculations were carried out for a range of angle of attack α from –20° to +25° and for a sideslip angle β from 0° to 20°. Based on the calculations carried out, analysis of the impact of the slip angle on the load on the stabilizers has been made.

Interoperability architecture for bridging computational tools: application to steel corrosion in concrete

A multiscale modelling framework, especially for corrosion modelling, requires not only robust computational tools but also an efficient datacentric architecture for handling information exchange at different modelling scales. Different computational solvers require and produce data in different programming languages and specific formats signifying a strong non-uniformity for an easy nexus with other solvers. This non-uniformity has created a need to focus on intermittent state-of-the-art datacentric software tools which aim to bridge data exchange heterogeneity across diverse set of solvers. Data organization in the form of metadata structures are presented as a standard for a coherent information representation regardless of the diverse nature of data formats specific to a scientific discipline. This fundamental work presents the concept, underlying terminology and working mechanism of a datacentric architecture tool SOFT5 for exchanging and interfacing data-flow between solvers and its present application to a concrete technology multiscale simulation network as a potential application.

Numerical Study of Micron-Scale Aluminum Particle Combustion in an Afterburner Using Two-Way Coupling CFD–DEM Approach

Metal additives have positive effects on the flow field such as increasing the temperature and thrust, while this may decrease local gaseous velocity due to inter-phase drag forces on the other hand. An in-house computational solver was developed to study the typical multiphase flow with combustion. The solver relies on Eulerian–Lagrangian model and the two-way coupling approach. Numerical simulations were carried out on an afterburning chamber of a solid-fuel ramjet to study the impact of using aluminum particles as metal additive. For micron-scale aluminum particles, the injection area, initial temperature, diameter and mass flow rate are varied influence factors, while the afterburning chamber and inlet primary high-temperature gases are fixed for all the test cases. It was found that better particle dispersion can result in higher combustion efficiency due to increasing the gas-particle contact area as well as better heat transfer and diffusion. Injecting aluminum particles with higher initial temperature and smaller diameter tend to get ignited faster and then burn more easily, which means releasing more heat in the length-limited flow field. However, the aluminum particle massflow rate has much more complicated comprehensive influence than the changing rules of the adiabatic combustion temperature based on the NASA CEA code.

Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM)” for the efficient and exact computation of 1st- and 2nd-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (i.e., model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2nd-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1st-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2nd-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (N2/2 + 3 N/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2nd-LASS requires very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1st-LASS and 2nd-LASS requires the same computational solvers as needed for solving (i.e., “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1st-LASS and the 2nd-LASS. Since neither the 1st-LASS nor the 2nd-LASS involves any differentials of the operators underlying the original system, the 1st-LASS is designated as a “first-level” (as opposed to a “first-order”) adjoint sensitivity system, while the 2nd-LASS is designated as a “second-level” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2nd-LASS that involve the imprecisely known boundary parameters. Notably, the 2nd-LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.

Reference Solutions for Time Domain Electromagnetic Solvers

In this study, a highly precise time domain solution of electromagnetic fields for a canonical structure is derived. Our reference solution is extremely useful for researchers, engineers, and developers to evaluate the accuracy of their computational results using commercial software or their self-developed codes. Rigorous solutions of a cylinder or sphere, which consists of a homogeneous medium, are derived in the complex frequency domain; they are numerically transformed into the time domain using fast inversion of Laplace transform. In addition, the field distribution at the desired specific observation time can be easily obtained. Furthermore, the numerical accuracy of the computational electromagnetic solvers is evaluated.

A More Configurable Finite Memory Event Triggering Control Scheme for Networked Control Systems

For a networked control system (NCS), this article proposes a more configurable event triggering control scheme that employs a finite memory structure with tuning parameters for efficiently utilizing network resources in transient and steady-state responses. The proposed finite memory event triggering control scheme (FMETS) uses the current and past states on the recent finite time horizon to asymptotically stabilize a closed-loop system. Tuning parameters can be configured according to the forgetting level, the triggering threshold, and the memory size. Since the FMETS takes a sample-and-hold approach, the undesirable Zeno effect can be avoided automatically. To use an efficient computation solver, the proposed FMETS is formulated into linear matrix inequalities (LMIs). It is shown through a simulation study that the proposed FMETS provides less conservative results and furthermore enables effective configuration for high control performance and efficient network resource utilization.

A new heuristic computational solver for nonlinear singular Thomas–Fermi system using evolutionary optimized cubic splines

In the present work, a new stochastic computing technique based on evolutionary cubic spline method (CSM) is introduced for solving nonlinear singular Thomas–Fermi system arising in atomic physics. The concept of cubic splines interpolation is engaged with an evolutionary optimization technique based on genetic algorithms (GAs) hybrid with sequential quadratic programming (SQP) to develop a proposed methodology, CSM-GASQP, that can solve nonlinear differential equations, and GA produces the optimized value for the coefficients of cubic splines, while SQP is used for rapid local refinements. The developed method CSM-GASQP for different lengths of the splines is applied effectively to solve the Thomas–Fermi equation for number of scenarios. Results show that proposed evolutionary paradigm CSM-GASQP is an effective, alternate, accurate, and reliable stochastic numerical solver for stiff nonlinear singular Thomas–Fermi systems.

Propulsion performance investigation of bio-inspired nano rotor base on fluid–structure interaction

In this paper, the bio-inspired blade motion is introduced to improve the propulsive performance of nano rotor at an ultra-low Reynolds number. However, the complex flow interacts with the flexible composite blade structure resulting in the change of nano rotor propulsion performance and the vibration of blade structure. A composite nano rotor with blade-pitch motion is investigated computationally with a computational solvers based on fluid–structure interaction. The finite element model for the composite rotor is created and verified with a non-contact modal test. It is found that the simulation results matched well with the experimental results. Successively, the propulsive performance of a rigid nano rotor is studied. The propulsive performance of the nano rotor is analysed at different bio-inspired pitch frequencies. The results show that the figure of merit of the bio-inspired pitch rotor increases because of the bio-inspired blade pitch motion. And it is also found that the improvement of the propulsive performance of the nano rotor varies with the pitch frequency. The propulsive performance of the flexible bio-inspired nano rotor is also studied with by using fluid–structure interaction method. It is found that the computational results for flexible nano rotor are lower than that for rigid nano rotor. It is evident that it is necessary to consider the flexibility of the composite nano rotor when investigating the propulsion performance of bio-inspired nano rotor. And the response of blade structure is also studied. Structural dynamic analysis shows that the blade structure vibrates with small amplitude. And two peak values are found at the rotation frequency and the fundamental frequency of the nano rotor structure.

Combined Effects of Thermal Non-equilibrium and Chemical Reactions on Hypersonic Air Flows Around An Orbital Reentry Vehicle

The present work computationally investigated the combined effects of thermal non-equilibrium and chemical reactions on hypersonic air flows around an orbital reentry vehicle during its reentry. The chemically reacting gas flow around the orbital vehicle was simulated for actual reentry trajectories, with a computational solver based on the Navier–Stokes–Fourier equations. Hypersonic flows for a wide range of flying altitudes from 40 to 80 km were examined. We first analyzed important properties such as Mach number, trans-rotational temperature, vibrational temperature, chemical species composition, and surface properties such as the maximum heat flux for various altitudes. We then investigated the distribution of heat flux on the surface of the orbital reentry vehicle and the coupled effects of thermal non-equilibrium and chemical reactions on the hypersonic air flows. The computed heat flux results were compared with actual flight test data obtained during reentry of the orbital vehicle. The computed results were found to be in excellent agreement with the flight test data until an altitude of 60 km. Finally, we propose an explanation why heat flux was over-predicted around 70 km altitude but returned to better agreement at the higher altitude of 80 km. The phenomenon is based on two competing effects: the shortcoming of the first-order law of heat flux in the NSF equations and the inaccurate description of the effects of thermal non-equilibrium on chemical reactions in Park’s two temperature model.