Steady-state Problem Research Articles

Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

Analytical Solutions of Heat Conduction Problems for Anisotropic Solid Body

Exact analytical solutions of heat conduction problems in anisotropic twodimensional solid bodies are presented in this study. Timedependent and steadystate problems are considered. A linear coordinate transformation is introduced which reduces the anisotropic heat conduction problem to an equivalent isotropic one. The solution of the anisotropic heat conduction problem is expressed in terms of solutions of the corresponding isotropic heat conduction problem. The connection of data of the applied linear coordinate transformation and the thermal material properties of anisotropic solid body is analysed. All result of the paper is based on the Fourier’s theory of heat conduction in solid bodies. Examples illustrate the applications of the developed method

Diffusive predator-prey models with fear effect in spatially heterogeneous environment

This article concerns diffusive predator-prey models incorporating the cost of fear and environmental heterogeneity. Under homogeneous Neumann boundary conditions, we establish the uniform boundedness of global solutions and global stability of the trivial and semi-trivial solutions for the parabolic system. For the corresponding steady-state problem, we obtain sufficient conditions for the existence of positive steady states, and then study the effects of functional responses and the cost of fear on the existence, stability and number of positive steady states. We also discuss the effects of spatial heterogeneity and spatial diffusion on the dynamic behavior and establish asymptotic profiles of positive steady states as the diffusion rate of prey or predator individuals approaches zero or infinity. Our theoretical results suggest that fear plays a very important role in determining the dynamic behavior of the models, and it is necessary to revisit existing predator-prey models by incorporating the cost of fear.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/70/abstr.html

Approximation-free-based adaptive integral robust control with prescribed performance for unknown pure feedback system with disturbance

This paper studies the application of prescribed performance control (PPC) method in digital control systems. First of all, the cause of the steady-state oscillation in PPC method is studied, and the influence of prescribed performance function (PPF) is analyzed. Then, an adaptive proportional-like control scheme with integrals is proposed. The adaptive gain is employed to tackle the steady-state oscillation problem, and the integral of transformation error is adopted to reject unknown disturbances and enhance the robustness of the controller. In the proposed control scheme, approximation frameworks are dispensable, and the asymptotic convergence of the tracking error is guaranteed by using integrals. Finally, the effectiveness and superiority of the proposed control strategy are verified through numerical simulations.

Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method

PurposeThe purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.Design/methodology/approachThe mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.FindingsThe numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.Originality/valueFinally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

Control of port-Hamiltonian systems with minimal energy supply

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.

Existence, Comparison, and Convergence Results for a Class of Elliptic Hemivariational Inequalities

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

Higher Order Multiscale Finite Element Method for Heat Transfer Modeling.

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

Spatial Decay of the Vorticity Field of Time-Periodic Viscous Flow Past a Body

We study the asymptotic spatial behavior of the vorticity field, $\omega(x,t)$, associated to a time-periodic Navier-Stokes flow past a body, $\mathscr B$, in the class of weak solutions satisfying a Serrin-like condition. We show that, outside the wake region, $\mathcal R$, $\omega$ decays pointwise at an exponential rate, uniformly in time. Moreover, denoting by $\bar{\omega}$ its time-average over a period and by $\omega_P:=\omega-\bar{\omega}$ its purely periodic component, we prove that inside $\mathcal R$, $\bar{\omega}$ has the same algebraic decay as that known for the associated steady-state problem, whereas $\omega_P$ decays even faster, uniformly in time. This implies, in particular, that "sufficiently far" from $\mathscr B$, $\omega(x,t)$ behaves like the vorticity field of the corresponding steady-state problem.

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation

The need for multiple interactive, real-time simulations using different parameter values has driven the design of fast numerical algorithms with certifiable accuracies. The reduced basis method (RBM) presents itself as such an option. RBM features a mathematically rigorous error estimator which drives the construction of a low-dimensional subspace. A surrogate solution is then sought in this low-dimensional space approximating the parameter-induced high fidelity solution manifold. However when the system is nonlinear or its parameter dependence nonaffine, this efficiency gain degrades tremendously, an inherent drawback of the application of the empirical interpolation method (EIM).In this paper, we augment and extend the EIM approach as a direct solver, as opposed to an assistant, for solving nonlinear partial differential equations on the reduced level. The resulting method, called Reduced Over-Collocation method (ROC), is stable and capable of avoiding the efficiency degradation. Two critical ingredients of the scheme are collocation at about twice as many locations as the number of basis elements for the reduced approximation space, and an efficient error indicator for the strategic building of the reduced solution space. The latter, the main contribution of this paper, results from an adaptive hyper reduction of the residuals for the reduced solution. Together, these two ingredients render the proposed R2-ROC scheme both offline- and online-efficient. A distinctive feature is that the efficiency degradation appearing in traditional RBM approaches that utilize EIM for nonlinear and nonaffine problems is circumvented, both in the offline and online stages. Numerical tests on different families of time-dependent and steady-state nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC and its superior stability performance.

A Benders Decomposition Based Algorithm for Steady-State Dispatch Problem in an Integrated Electricity-Gas System

Optimally operating an integrated electricity-gas system (IEGS) is significant for the energy sector. However, the IEGS operation model's nonconvexity makes it challenging to solve the optimal dispatch problem in the IEGS. This letter proposes a new Benders decomposition-based (IBD) algorithm catering to a commonly used steady-state dispatch model of the IEGS. This IBD algorithm leverages a refined decomposition structure where the subproblems are linear and ready to be solved in parallel. We analytically compare our IBD algorithm with an existing Benders decomposition algorithm and a typical piecewise linearization method. Case studies have substantiated the higher computational efficiency of our IBD algorithm.

Numerical study of a one and two-dimensional heat flow using finite volume

In this paper, the Finite Volume numerical scheme has been used to solve one-dimensional unsteady state and two-dimensional steady-state heat flow problems with the initial condition and Dirichlet boundary condition. Mainly two schemes are used to compute the numerical solutions of unsteady-state flow and one scheme for steady-state flow. The applicability of these schemes is illustrated using simple test problems. The numerical solutions obtained by these schemes are compared with the analytic solutions to check the accuracy of the developed scheme.

Assessment of confined aquifer response to recharge variations and water inflow distributions using analytical approach.

Quantification of the amount of the exchanged water between the surface water and a confined aquifer is a basic step in water balance and environmental hydraulics. The hydraulic connection between a surface water and a confined aquifer may occur through different recharge variations. The current research presents new analytical solutions for confined aquifer response to recharge variations and different inflow distributions. Different cases were studied, where a constant piezometric head is applied at the right boundary of the 2D confined aquifer plane and various distributions of water inflow through the recharging windows are considered on a part and/or parts of the left boundary. Finally, a uniform water inflow distribution on parts of the left boundary and a uniform distribution of water outflow at the right boundary was considered. Both steady and unsteady state problems can be solved using proposed equations. The performance of developed analytical solutions was examined compared to the numerical finite difference modeling. The results show reasonable precision of the developed analytical solutions. The developed solutions can be used as a benchmark to verify numerical approaches with similar boundary conditions.

Multi-objective steady-state optimization for a complex bioprocess in glycerol metabolism

This paper addresses the multi-objective steady-state optimization of a complex nonlinear system in glycerol metabolism. In the bioconversion system, we consider some mechanisms including the regulation for dha regulon, enzyme-catalytic kinetics, the inhibition of intracellular 3-hydroxypropionaldehyde (3-HPA), and so on. Firstly, we propose several new multi-objective steady-state optimization problems. These problems can simultaneously optimize any two of the following objectives: (1) minimization of the intracellular 3-HPA concentration; (2) maximization of the production rate of extracellular 1,3-propanediol (1,3-PDO); (3) maximization of the conversion rate of extracellular glycerol to extracellular 1,3-PDO; (4) maximization of the conversion rate of extracellular glycerol. Then a hybrid optimization method is proposed to solve the presented complex multi-objective problems. This optimization method couples a deterministic interior point approach and a stochastic genetic algorithm in the framework of normal-boundary intersection (NBI) method. After solving the proposed multi-objective problems of the addressed bioconversion system, we finally obtain their approximate Pareto optimal sets. The analysis for the attained optimization results is given. And a comparative study is also presented.

Probability analysis of steady-state voltage stability considering correlated stochastic variables

The penetration of renewable generation, such as photovoltaic and wind power in power networks will significantly impact the system’s stability in future, due to randomness and intermittency in the natural environment. Moreover, the correlations between renewable generation may enhance the impact on power system's stability. In this paper, cumulant-based maximum entropy method (CMEM) combined with Nataf Transform (NT) is proposed to analyze the steady-state voltage stability problem considering correlations and uncertainties of power injections and consumptions. The proposed methodology is tested on the IEEE 30-node and IEEE 57-node test systems. Taking the results of Monte Carlo method (MCM) as the experimental control group, this paper compares the proposed method with series expansion method (SEM), discusses and analyzes their calculation results and speed under different scenarios, and the results of the comparison prove the validity, accuracy and rapidity of the CMEM.

A new fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow

A new fifth-order alternative finite difference multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve the hyperbolic conservation laws in this paper. With the application of the original finite difference multi-resolution WENO schemes (Zhu and Shu, 2018; Zhu and Shu, 2020), a series of unequal-sized central spatial stencils are adopted to perform the WENO procedures, but the difference is that the methodology adopted in this paper is directly based on the point values of the solution rather than on the flux values. The proposed new multi-resolution WENO scheme can inherit many advantages of the original high order schemes, such as the arbitrary choice of the linear weights, the maintenance of the essentially non-oscillatory property in the vicinity of strong discontinuities, the smaller L1 and L∞ truncation errors than that of the same order classical WENOJS schemes (Balsara et al., 2016; Jiang and Shu, 1996) in smooth regions, and better convergence properties of the residue when solving some steady-state problems. Compared with the original multi-resolution WENO schemes, the presented WENO scheme also has some advantages. Firstly, an arbitrary monotone flux can be used in this framework, while the original method is only suitable for smooth flux splitting technique. Secondly, it has smaller L1 and L∞ truncation errors than that of the original same order multi-resolution WENO scheme. And finally, it avoids some time-consuming intermediate processes in the original multi-resolution WENO reconstruction procedures, thus resulting in less calculation time under the same conditions. Some inviscid and viscous numerical examples are provided to verify the superior performance of the new fifth-order alternative finite difference multi-resolution WENO scheme.

Verification of ansys and matlab Heat Conduction Results Using an “Intrinsically” Verified Exact Analytical Solution

Abstract A two-dimensional (2D) transient thermal conduction problem is examined, and numerical solutions to the problem generated by ansys and matlab, employing the finite element (FE) method, are compared against an “intrinsically” verified analytical solution. Various grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ansys software, including coarse, medium, and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body, and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. Two different test cases are examined for the various numerical solutions using selected grid densities. The first case involves uniform constant heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case still involves uniform constant heating but for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times for both ansys and matlab are extremely small, the errors found within the short-duration test are more significant, in particular when the heating is suddenly set “on.” Surprisingly, very small errors occur when the heating is suddenly set “off.” The solution developed using the matlab differential equation solver is found to have errors an order of magnitude larger than those generated using ansys with a similar mesh size and same FE type (quadratic triangular). This occurs not only during the transient but also for steady-state problems. However, the matlab computational efficiency is superior. Also, when using ansys, at early times the numerical errors are very sensitive to the time-step chosen rather than the spatial discretization. Therefore, the time-steps have to be very small near a singularity.

Exact solutions of steady-state dynamic responses of a laminated composite double-beam system interconnected by a viscoelastic layer in hygrothermal environments

The current paper is the first to study the steady-state dynamic problem of a laminated composite double-beam system (LCDBS) under hygrothermal environments, in a systematic manner. The LCDBS consists of two laminated composite beams connected by a uniformly distributed viscoelastic layer and is supported on the Winkler-Pasternak elastic foundation. The governing equations of LCDBS are derived based on the Euler–Bernoulli beam hypothesis considering the free expansion caused by temperature variation and moisture concentration. Exact solutions for steady-state dynamic responses of the LCDBS are obtained by employing the Green's functions method and superposition principle. In the meantime, the implicit equation calculating the natural frequency of the LCDBS is proposed. The material properties are considered to be temperature- and moisture-dependent. Numerical calculations are performed to explore the influences of physical parameters of the connecting layer, foundation stiffness, fiber orientation angle, temperature variation, and moisture concentration on steady-state dynamic responses of the LCDBS. Outcomes reveal that the bond between the two beams can be strengthened by increasing the stiffness and damping coefficient of the connecting layer, and the increase of the temperature variation can significantly reduce the system stiffness and enhance the dynamic responses of the two beams.

Use of Monte Carlo code MCS for multigroup cross section generation for fast reactor analysis

Multigroup cross section (MG XS) generation by the UNIST in-house Monte Carlo (MC) code MCS for fast reactor analysis using nodal diffusion codes is reported. The feasibility of the approach is quantified for two sodium fast reactors (SFRs) specified in the OECD/NEA SFR benchmark: a 1000 MWth metal-fueled SFR (MET-1000) and a 3600 MWth oxide-fueled SFR (MOX-3600). The accuracy of a few-group XSs generated by MCS is verified using another MC code, Serpent 2. The neutronic steady-state whole-core problem is analyzed using MCS/RAST-K with a 24-group XS set. Various core parameters of interest (core keff, power profiles, and reactivity feedback coefficients) are obtained using both MCS/RAST-K and MCS. A code-to-code comparison indicates excellent agreement between the nodal diffusion solution and stochastic solution; the error in the core keff is less than 110 pcm, the root-mean-square error of the power profiles is within 1.0%, and the error of the reactivity feedback coefficients is within three standard deviations. Furthermore, using the super-homogenization-corrected XSs improves the prediction accuracy of the control rod worth and power profiles with all rods in. Therefore, the results demonstrate that employing the MCS MG XSs for the nodal diffusion code is feasible for high-fidelity analyses of fast reactors.

High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical L2 projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.