Steady Problems Research Articles

An implicit function formulation for optimization of discretized index-1 differential algebraic systems

A formulation for the optimization of index-1 differential algebraic equation systems (DAEs) is described that uses implicit functions to remove algebraic variables and equations from the optimization problem. The formulation uses the implicit function theorem to calculate derivatives of functions that remain in the optimization problem in terms of a reduced space of variables, allowing it to be solved with second-order nonlinear optimization algorithms. The formulation is shown to lead to more reliable solver convergence when compared with a full-space simultaneous formulation in challenging case studies involving a chemical looping combustion reactor. In a steady state simulation problem, the implicit function formulation solves 82 out of 100 instances, while the full space formulation solves only 42 out of 100 instances. In a dynamic optimization problem, the implicit function formulation solves 189 out of 200 instances, while the full space formulation solves only 152 out of 200 instances.

On numerical solution stability of the Laplace equation in absence of the Dirichlet boundary condition: Benchmark problem proposal

A fundamental 1D steady state diffusion problem using only third type boundary condition also named as the Robin or Fourier type is analysed. In the heat transfer problems this boundary condition is called convective boundary condition using heat transfer coefficient ( α ) as a constant. The main idea of the proposed Benchmark problem is to test the solution accuracy in both limiting cases of α approaching to zero and to infinity. The infinite case limits to Dirichlet boundary condition and should not be the problem, while the zero limit leads to trivial solution of zeros in the absence of the Dirichlet boundary condition. The application of the Benchmark would be in freeze drying, where similar boundary conditions occur. The presented BEM numerical solution failed for α > 1 0 13 in double precision computation for two 2D formulations: cartesian and axisymmetric. The low limit results failed for α < 1 0 − 13 for cartesian formulation. This is expected, while the failure of axisymmetric formulation results already at moderate low α < 1 0 − 3 value is not. The solution stability and accuracy are found strong dependent to the fundamental solution integrals accuracy. The effect of system matrix preconditioner is also studied.

Heat Conduction Plate Layout Optimization Using Physics-Driven Convolutional Neural Networks

Optimizing heat conduction layout is essential during engineering design, especially for sensible thermal products. However, when the optimization algorithm iteratively evaluates different loading cases, the traditional numerical simulation methods usually lead to a substantial computational cost. To effectively reduce the computational effort, data-driven approaches are used to train a surrogate model as a mapping between the prescribed external loads and various geometry. However, the existing model is trained by data-driven methods, which require intensive training samples from numerical simulations and do not effectively solve the problem. Choosing the steady heat conduction problems as examples, this paper proposes a physics-driven convolutional neural networks (PD-CNNs) method to infer the physical field solutions for randomly varied loading cases. After that, the particle swarm optimization (PSO) algorithm is used to optimize the sizes, and the positions of the hole masks in the prescribed design domain and the average temperature value of the entire heat conduction field is minimized. The goal of reducing heat transfer is achieved. Compared with the existing data-driven approaches, the proposed PD-CNN optimization framework predicts field solutions that are highly consistent with conventional simulation results. However, the proposed method generates the solution space without pre-obtained training data. We obtained thermal intensity results for holes 1, hole 2, hole 3, and hole 4 with 0.3948, 0.007, 0.0044, and 0.3939, respectively, by optimization PD-CNN model.

A unified lattice Boltzmann framework for combined radiation-conduction heat transfer

As the wider application of combined radiation-conduction heat transfer, effective numerical framework is required to precisely predict the heat transfer process. In this paper, a unified lattice Boltzmann framework is proposed for combined radiation-conduction heat transfer problem. As a complete form of lattice Boltzmann framework, the radiative transfer equation (RTE) and energy equation are both solved under the versatile, consistent D1Q3, D2Q9 and D3Q15 lattice model to construct a combined LB framework. Through several numerical cases from 1-D to 3-D problems for both steady and transient problems, we validated the applicability and accuracy of our framework. Besides, taking account of various parameters, we discuss the effects of different medium properties and anisotropic scattering condition. Our unified LB framework for radiation-conduction transfer is tested to be available for muti-dimensional problems through 1-D to 3-D cases. Furthermore, the anisotropic scattering condition is considered as an extension of our framework. The present work is expected to provide an effective numerical tool for various of combined radiation-conduction heat transfer questions.

ADER scheme with a simplified solver for the generalized Riemann problem and an average ENO reconstruction procedure. Application to blood flow

We present numerical schemes for solving hyperbolic balance laws, admitting stiff source terms, to high-order of accuracy in both space and time. The schemes belong to the ADER family of methods and as such rest on two building blocks, namely a non-linear spatial reconstruction procedure and the solution of a generalized Riemann problem (GRP), in which the data is piece-wise smooth and the equations include source terms. This paper presents contributions on both building blocks. Concerning spatial reconstruction we present various versions of a new Essentially Non-Oscillatory (ENO) type reconstruction, called here Averaged ENO (AENO). As to the generalized Riemann problem, we present an improved solver based on two basic ingredients, namely the implicit Taylor series expansion reported in Toro and Montecino (2015) and the simplified Cauchy–Kowalewskaya procedure reported in Montecinos and Balsar (2020).The resulting ADER schemes are implemented and systematically assessed for the linear advection equation and for non-linear hyperbolic system that governs blood flow, with a tube law admitting arteries or veins. For the linear advection equation we implement the new schemes to accuracy ranging from first to seventh order; convergence rate studies confirm that the theoretically expected accuracy is achieved for most versions of the various schemes studied. For the non-linear system we implement the new schemes to accuracy ranging from first to fifth order. Convergence rate studies based on problems with smooth solutions confirm again that the methods attain the theoretically expected accuracy. For the non-linear system the methods are also assessed for Riemann problems for blood flow in arteries with exact solution containing smooth parts and discontinuities, elastic jumps (shocks), and contact discontinuities. Furthermore, for the non-linear system the methods are also assessed for blood flow in veins for a steady problem with smooth analytical solution. As a final assessment of the potential applicability of the methods for realistic simulations in haemodynamics, we apply the methods on a network of 37 blood vessels, for which experimental data is available. The new schemes presented in this paper are simpler than existing versions of ADER methods, involving implicit Taylor series and the Cauchy–Kowalewskaya procedure, published in the literature and, overall, the computational results demonstrate that the new schemes give comparable or superior results, with respect to existing high-order schemes.

Effects of curvature on triple flame propagation in fuel-oxidizer counterflow

In this study, we analyze the propagation of an expanding triple flame in an axisymmetric counterflow of a fuel against an oxidizer. The problem is formulated as a thermo-diffusive model with one-step global reaction. An asymptotic analysis in the limit of large activation energy and weak strain rate is conducted. The study is supported and complemented by numerical simulations carried out for arbitrary values of the strain rate. In addition to the flame front curvature 1/Rt associated with the variation in the reactants concentrations transverse to the mixing layer, the propagation of the expanding triple flame also depends on the azimuthal curvature 1/Rf where Rf is the front leading edge radial distance. As the triple flame expands to large radial distances, its propagation becomes quasi-steady. Under a quasi-steady state assumption, an explicit expression is derived for the displacement speed of the triple flame, which is found to be linearly proportional to the total curvature 1/Rf+1/Rt. Two-dimensional axisymmetric simulations are conducted to validate in particular the quasi-steady assumption. These include transient simulations of the expanding triple flame which are compared to the numerical solution of the steady eigenvalue problem obtained in a frame attached to the propagating front under a quasi-steady assumption. Following a transient ignition phase, the triple flame is found to propagate in a quasi-steady manner when Rf (measured with the stoichiometric planar flame thickness) exceeds 5, approximately. Although the theoretical analysis is performed in the weak strain limit, the linear dependence of the triple flame speed on the curvature 1/Rfis found to be applicable over a wide range of strain rates. Besides, the analysis is extended for inwardly propagating triple flames (flame holes) and similar expressions describing the relationship between displacement speed of the triple flame and curvatures are obtained.

The Finite Element Numerical Study of Transonic Flow of Moist Air with Nonequilibrium Condensation

The finite element numerical simulation of the transonic flow with condensation is studied in this paper. The transformed weak form is numerically discretized by using the classical Galerkin finite element spatial discretization scheme and the backward Euler difference time discretization scheme. Because the coupled Euler equations are convection equations, the results obtained by the classical Galerkin method often have nonphysical numerical oscillations, so the stabilization method is introduced. Another important feature of compressible Euler equation is the existence of shock wave. Because the thickness of shock wave is small and difficult to capture, the shock wave can be better captured by adding artificial viscosity term. Here, the streamline upwind/Petrov–Galerkin (SUPG) finite element method with good stability and the isotropic diffusion method with shock capture ability are used to solve the problem of transonic flow with condensation. In the simulation of steady problems, in order to improve the calculation accuracy of the existing area of the shock wave, the adaptive mesh refinement technology is added, which refines the mesh in the region of the shock wave to improve the resolution of the shock wave. Numerical experiments verify the feasibility and stabilization of the numerical method. Finally, the simulation of transonic flow around NACA-0012 and parallel-jet nozzle A1 has obtained the better numerical results.

Multilevel field-split preconditioners with domain decomposition for steady and unsteady flow problems

With the incremental applications of Newton–Krylov methods for solving large sparse nonlinear systems of equations, the design of robust and scalable linear preconditioners plays an essential role for the whole solver. In this paper, we investigate the family of field-split (FS) preconditioners with different combinations of physics-based and domain decomposition methods, applied to the two typical fluid problems, i.e., the unsteady flow through fractured porous media and the steady buoyancy driven flow. In the implementation, several new versions of FS preconditioners are considered under the framework of the domain decomposition technique: additive FS, multiplicative FS, Schur-complement FS, and the constrained pressure residual (CPR) method, where the inverse of corresponding matrices is approximated by using the restricted additive Schwarz (RAS) algorithm. Rigorous eigenvalue analysis for various FS preconditioners is also provided for facilitating the design of algorithms. In particular, our approach further enhances the numerical performance by presenting a family of multilevel field-split methods for efficiently preconditioning. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the proposed preconditioning strategies for both standard benchmarks as well as realistic flow problems on a supercomputer.

Comparative Study of Analytic and Numerical Solutions of Steady-state Temperature Distribution on Semi-circular Plate Using Laplace Equation

In this paper numerical methods have been used to solve two dimensional steady state heat flow problem in polar coordinates with Dirichlet boundary conditions inside a semi-circular plate and the work focuses on the numerical methods for solving Laplace equation; finite difference schemes and Gauss elimination method. The numerical solution is compared with exact solution of the same problem. Finally, we analyze the absolute error in different number of iterations to check the accuracy of schemes.

Dual solutions for MHD hybrid nanofluid stagnation point flow due to a radially shrinking disk with convective boundary condition

PurposeThis paper aims to study the stagnation point flow of Al2O3–Cu/H2O hybrid nanofluid over a radially shrinking disk with the imposition of the magnetic field, viscous-Ohmic dissipation and convective boundary condition.Design/methodology/approachSimilarity variables are introduced and used in reducing the governing partial differential equations into a system of ordinary differential equations. A built-in bvp4c solver in MATLAB is then used in the computation of the numerical solutions for equations (7) and (8) subject to the boundary conditions (9). Then, the behavior of the flow and thermal fields of the hybrid nanofluid, with various values of controlling parameters, are analyzed.FindingsThe steady flow problem resulted in multiple (dual) solutions. A stability analysis performed to identify the stable solution applicable in practice revealed that the first solution is stable while the second solution is unstable. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than the Al2O3–H2O nanofluid. Thus, the hybrid nanofluid has a better heat transfer performance than the nanofluid. Besides that, the presence of the magnetic field, suction, convective boundary condition and the enhancement of nanoparticle volume fraction of Cu augments the skin friction coefficient and Nusselt number of the hybrid nanofluid. Meanwhile, the presence of viscous-Ohmic dissipation reduces the heat transfer performance of the fluid.Originality/valueTo the best of the authors’ knowledge, the present results are original and new for the study of the flow and heat transfer of Al2O3–Cu/H2O hybrid nanofluid past a permeable radially shrinking disk. Considerable efforts have been directed toward the study of the boundary layer flow and heat transfer over stretching/shrinking surfaces and disks because of its numerous industrial applications, such as electronic, power, manufacturing, aerospace and transportation industries. Common heat transfer fluids such as water, alumina, cuprum and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals have higher thermal conductivities than these fluids. Therefore, it is desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid but has higher heat transfer properties.

Discretization Corrected Particle Strength Exchange for Steady State Linear Elasticity

Discretization corrected particle strength exchange (DC PSE) is a particle based spatial differential operator designed to solve meshless continuum mechanics problems. DC PSE is a spatial gradient operator that can discretize a computational domain with randomly distributed particles, provided that each particle has enough neighboring particles. In contrast, conventional methods such as the standard finite difference method require the computational domain to be discretized into a Cartesian grid. In linear elasticity simulations, especially steady state cases, this domain is mostly discretized using mesh-based methods such as finite element. However, while particle methods such as smoothed particle hydrodynamics (SPH) have been widely applied to solve dynamic elasticity problems, they have rarely been used in steady state simulations. In this study, a DC PSE operator was used to solve steady linear elasticity problems in a two-dimensional domain. The result of the DC PSE numerical simulation was compared to numerical results, empirical formula results, and results from conventional commercial finite element software, respectively.

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

On the dissipation of conforming and discontinuous Galerkin schemes for the incompressible Navier–Stokes equations

In this paper, we improve the numerical performance of the classical conforming finite element schemes for the time-dependent incompressible Navier–Stokes equations by adding dissipation. This is a physics-inspired approach, and the dissipative terms are constructed through the discontinuity of numerical quantities across interior edges and, therefore, decouple the space and time discretizations when compared with the streamline-upwind Petrov–Galerkin for the time-marching methods. In particular, the order of h (edge diameter) in the dissipative terms is motivated by the energy stability and error equation associated with the unsteady problem. Furthermore, we point out that the added dissipation may also be viewed as an alternative for the grad-div stabilization from the physical approach in the unsteady problem. The added dissipation is naturally within the framework of the variational multiscale and thus could serve as implicit subgrid-scale models in large eddy simulations. Numerical experiments with a jump of the gradient are performed. In addition, we test the ideas with the discontinuous Galerkin formulations. Numerical results indicate that our suggested dissipation is helpful in reducing numerical errors and is competitive when compared with other conventional stabilization available in the literature. Finally, we show that the changes in the physical role of the same terms may significantly change their corresponding numerical behaviors through examples on the steady problems.

A new method for the evaluation of the vacuum boundary in circular and D-shaped Tokamaks

A new method for the evaluation of the vacuum boundary in a Tokamak is introduced. It is a two-step iterative algorithm that converges to the actual value of the boundary location only after a few iterations. The key point in this method is to solve a well-posed boundary value problem that satisfies a set of exact boundary conditions. A Cauchy problem in steady heat-conduction problem is used to explain the method first. Then, inverse evaluation of the vacuum boundary location in circular as well as D-shaped Tokamaks are considered. For circular Tokamak, the method does not require any regularization. The algorithm is quite simple and very effective. Numerical examples are presented to study the performance of the algorithm in the presence of noise.

Assessment and numerical validation of a normal mode stability analysis for conjugate heat transfer

This paper aims to evaluate and validate a mathematical coupling model, based on a normal mode stability analysis, for steady conjugate heat transfer problems. In order to achieve this goal, an aerothermal computational code was designed specifically ex novo to meet the needs of the current study. A large number of numerical simulations were carried out within the framework of the Dirichlet-Robin fluid-solid interface procedure and with conditions strictly identical to those adopted in the mathematical model. These coupled computations have been synthesized in five representative cases. The computed results show that the stability limit and the numerical Biot number derived from the coupling model are very reliable parameters. Many other physical and numerical aspects, such as the splitting of the coupling period are also considered in this work from which some interesting and promising results are obtained. At the end of this paper, we assess the general potential, limits and constraints of the mathematical model and its extension to multidimensional cases is discussed.

Entropy Generation and Natural Convection Heat Transfer of (MWCNT/SWCNT) Nanoparticles around Two Spaced Spheres over Inclined Plates: Numerical Study

A numerical study is conducted to evaluate the steady natural convective heat transfer problem and entropy generation of both single wall (SWCNT) and multi wall (MWCNT) nanoparticles with water as a base liquid over two spaced spheres. The isothermally heated spheres are located between two plates of short length. The cooled plates are maintained at different inclination angles. A numerical approach based on the finite volume method and multigrid acceleration was used to solve the governing equations. The effects of nanoparticle type, volume fraction, the inclination angle of the plates and the Rayleigh numbers are well-considered. Results reveal that there is a remarkable enhancement of the average Nusselt number over the plates for MWCNT nanoparticles with 63.15% from the inclination angle 0° to 30°. Furthermore, optimal heat transfer rates over the plates for MWCNT nanoparticles equates to 1.9, which is obtained for the inclination 30° and a Rayleigh number of 106. However, for SWCNT nanoparticles, the same equates 0.9, which is obtained for the inclination 90° and a Rayleigh number of 106. The comprehensive analysis is presented under some well-defined assumptions which show the reliability of the present investigation.

An arbitrary Lagrangian-Eulerian SPH-MLS method for the computation of compressible viscous flows

In this work we present a high-accurate discretization to solve the compressible Navier-Stokes equations using an Arbitrary Lagrangian-Eulerian meshless method (SPH-MLS), which can be seen as a general formulation that includes some well-known meshfree methods as a particular case. The formulation is based on the use of Moving Least Squares (MLS) approximants as weight functions on a Galerkin formulation and to accurate discretize the convective and viscous fluxes. This formulation also verifies the discrete partition of unity and reproduces the zero-gradient condition for constant functions. Convective fluxes are discretized using Riemann solvers. In order to obtain high accuracy MLS is also used for the high-order reconstruction of the Riemann states.The accuracy and performance of the proposed method is demonstrated by solving different steady and unsteady benchmark problems.

Robust RRE technique for increasing the order of accuracy of SPH numerical solutions

This study presents the use of a post-processing technique called repeated Richardson extrapolation (RRE) to improve the accuracy of numerical solutions of local and global variables obtained using the smoothed particle hydrodynamics (SPH) method. The investigation focuses on both the steady and unsteady one-dimensional heat conduction problems with Dirichlet boundary conditions, but this technique is applicable to multidimensional and other mathematical models. By using all the variables of the real type and quadruple precision (extended precision or Real*16) we were able to, for example, reduce the discretization error from 1.67E−08 to 3.46E−33 with four extrapolations, limited only by the round-off error and, consequently, determining benchmark solutions for the variable of interest ψ(1/2) using the SPH method. The increase in CPU time and memory usage owing to post-processing was almost null. RRE has proven to be robust in determining up to a sixteenth order of accuracy in meshless discretization for the spatial domain.

An ALE residual distribution scheme for the unsteady Euler equations over triangular grids with local mesh adaptation

This work presents a novel interpolation-free mesh adaptation technique for the Euler equations within the arbitrary Lagrangian–Eulerian framework. For the spatial discretization, we consider a residual distribution scheme, which provides a pretty simple way to achieve high order accuracy on unstructured grids. Thanks to a special interpretation of the mesh connectivity changes as a series of fictitious continuous deformations, we can enforce by construction the so-called geometric conservation law, which helps to avoid spurious oscillations while solving the governing equations over dynamic domains. This strategy preserves the numerical properties of the underlying, fixed-connectivity scheme, such as conservativeness and stability, as it avoids an explicit interpolation of the solution between different grids. The proposed approach is validated through the two-dimensional simulations of steady and unsteady flow problems over unstructured grids.

Spatial parallel disturbance region update method with OpenMP for steady compressible flows

We present the spatial parallelization of the disturbance region update method (DRUM) with OpenMP, named the spatial parallel disturbance region update method (spDRUM). DRUM is an acceleration methodology for compressible flows, employing dynamic computational domains (DCDs) to eliminate the worthless computational effort of the conventional global-update method. However, the DCDs may cause a load-imbalance issue, making DRUM difficult to execute efficiently in parallel. In this work, parallel strategies of steps in spDRUM are introduced, and a dynamic load balancer that is established based on the equivalent number of cells in the DCDs and a combined grid partitioning method is proposed to exploit the full acceleration potential of spDRUM. Besides, a concept of sub-blocks that takes advantage of the memory-shared characteristic of OpenMP is embedded in the data structure to simplify the inner-boundary treatment. Numerical test cases demonstrate that spDRUM accomplishes remarkable convergence speed for solving all steady compressible flow problems; benefiting from the coupled contribution of parallelism and DCDs, the total speed-up ratio of spDRUM could be significantly greater than that of the ideal case ignoring parallel overheads and that of the global-update method with the same number of threads; the proposed grid partitioning method could generate various types of partitions efficiently and flexibly, keeping load balanced for multiblock structured grids; embedding the concept of sub-blocks in the data structure makes the number of inner-boundary cells unchanged and remains optimal, regardless of the variation in partitions.