Efficient Numerical Method For Solution Research Articles

An efficient numerical method for modeling silver powder heat exchanger in dilution refrigerator

Dilution refrigerator provides continuous ultralow temperature environments as low as 10 mK. It has been widely used in a variety of important applications such as quantum computations. The silver powder heat exchanger in a dilution refrigerator plays a crucial role in realizing ultralow temperatures on the order of 10 mK by precooling the circulation of helium mixtures. To study the silver powder heat exchanger quantitatively, we have proposed an efficient numerical solution method for its thermodynamic model. This method utilizes constraint conditions cleverly to convert the initial value problem of differential equations into a boundary value problem, allowing us to solve the control equations using the existing ODE function quickly. Furthermore, we demonstrate the application of this method in the evaluation and design of silver powder heat exchanger. The research results of this paper have certain significance for the development of dilution refrigerator.

МАТЕМАТИЧНА МОДЕЛЬ ВЗАЄМОДІЇ СТАЦІОНАРНИХ SH-ХВИЛЬ З СИСТЕМОЮ КРИВОЛІНІЙНИХ ТРІЩИН У НАПІВПРОСТОРІ

A method for solving mathematical physics problems is proposed for semi-infinite media containing systems of curved crack-slits. Most of the studies known from the literature relate to the problems of diffraction of elastic waves on straight and circular cuts. However, in reality, the crack is usually not straight or circular. Studies have shown that the curvature of a crack significantly affects the value of the dynamic stress intensity coefficients. The value of this parameter also depends on the proximity of the defects to each other, since they always fall within the range of the reflected wave. The stress-strain state of media with complex properties can be effectively modeled by computing complexes in combination with software systems. Most studies are devoted to the development of the finite element method. However, the method of integral equations is very effective for solving anti-planar problems of diffraction theory. The advantage of this method is the reduction in the number of spatial variables, the high speed of convergence, and the possibility of using various efficient numerical solution methods. The method also has the ability to build efficient parallel computing schemes. A unified approach to solving the problem is developed on the basis of singular integral equations (SIEs). The corresponding dynamic boundary value problems for a clamped and force-free half-plane are investigated. The influence of the defect curvature, their interaction, and the proximity of the boundary on the magnitude and nature of the dynamic stress intensity coefficients is studied. Parallel algorithms allow to significantly reduce the computation time and analyse the characteristics of the wave field in more detail. The combination of the SIE method, which reduces the dimensionality of the problem by one, and provides significant savings in computing time due to the parallelization of computational procedures, leads to a significant increase in the efficiency of the proposed algorithm. The method can be used to assess the influence of various mechanical or geometric factors on the strength of bodies with defects.

Wellbore multiphase flow behaviors of gas kick in deep water horizontal drilling

During the deepwater drilling, the complicated gas-liquid-solid multiphase flow will occur if the formation gas enters and migrates in the wellbore. Through understanding of the wellbore flow behaviors is of great importance for the blowout prevention and well control. Considering the dynamic mass and heat transfer process in wellbore caused by alternating ambient temperature field, a multiphase flow model of multicomponent fluid in wellbore is deduced and developed, including the continuity equation, momentum conservation equation and energy conservation equation. Furthermore, the corresponding initial and boundary conditions are proposed for different working conditions in deepwater drilling, and an efficient numerical solution method is established, including dynamic mesh generation method and discrete solution method of partial differential equations. Applied in a deep-water kicking well, the proposed model is used to analyze the multiphase flow rules in the wellbore. The results show that in the process of annular fluid returning from the bottomhole, the pressure generally decreases linearly, while the temperature change is nonlinear. The temperature first rises and then falls at the formation section, and first falls and then rises at the seawater section. Furthermore, the pit gain increases approximately in a quadratic polynomial relationship, caused by the rise and expansion of gas in the wellbore, and the pressure drop and gas influx rate increase at the bottomhole. In the process of kick evolution, the standpipe pressure and bottomhole pressure gradually decrease, which can be an important sign for kick detection.

Investigations of Couette Flow Unsteady Radiative Convective Heat Transfer in a Vertical Channel Using the Generalized Method of Lines (MOL)

This study describes a very efficient and fast numerical solution method for the non-steady free convection flow with radiation of a viscous fluid between two infinite vertical parallel walls. The method of lines (MOL) is used together with the Runge-Kutta ODE Matlab solver to investigate this problem numerically. The presence of radiation adds more stiffness and numerical complexity to the problem. A complete derivation in dimensionless form of the governing equations for momentum and energy is also included. A constant heat flux condition is applied at the left wall and a transient numerical solution is obtained for different values of the radiation parameter (<i>R</i>). The results are presented for dimensionless velocity, dimensionless temperature and Nusselt number for different values of the Prandtl number (<i>Pr</i>), Grashof number (<i>Gr</i>), and the radiation parameter (<i>R</i>). As expected, the results show that the convection heat transfer is high when the Nusselt number is high and the radiation parameter is low. It is also shown that the solution method used is simple and efficient and could be easily adopted to solve more complex problems.

An efficient numerical solution method for detailed modelling of large 5th generation district heating and cooling networks

Numerical simulations of district heating and cooling networks constitute a valuable tool for system design and optimization. This paper presents a novel thermo-hydraulic network model, which addresses particularly large 5th generation district heating and cooling (5GDHC) networks. We take into account the current flow regime for heat loss calculation, which significantly improves accuracy during zero mass flux compared to a widely used literature model. We demonstrate our model's capability on a real-world 5GDHC network with 6 km length and 180 distributed heat pumps. In contrast to existing models, we take into account the temperature dependency of the viscosity, which leads to a deviation of up to 65% in pressure drop in the given example. Yet, the performance of our implementation is considerably higher in comparison to simplified models of similar size that use generic equation solver, such as Modelica, as we avoid typical performance issues due to tight coupling and solution of large equation systems. The model is implemented using error-controlled, adaptive time step solver CVODE and made available as open-source project for reference and general usage.

An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD

The magnetohydrodynamics (MHD) equations are continuum models used in the study of a wide range of plasma physics systems, including the evolution of complex plasma dynamics in tokamak disruptions. However, efficient numerical solution methods for MHD are extremely challenging due to disparate time and length scales, strong hyperbolic phenomena, and nonlinearity. Therefore the development of scalable, implicit MHD algorithms and high-resolution adaptive mesh refinement strategies is of considerable importance. In this work, we develop a high-order stabilized finite-element algorithm for the reduced visco-resistive MHD equations based on the MFEM finite element library (mfem.org). The scheme is fully implicit, solved with the Jacobian-free Newton-Krylov (JFNK) method with a physics-based preconditioning strategy. Our preconditioning strategy is a generalization of the physics-based preconditioning methods in Chacón et al. (2002) [3] to adaptive, stabilized finite elements. Algebraic multigrid methods are used to invert sub-block operators to achieve scalability. A parallel adaptive mesh refinement scheme with dynamic load-balancing is implemented to efficiently resolve the multi-scale spatial features of the system. Our implementation uses the MFEM framework, which provides arbitrary-order polynomials and flexible adaptive conforming and non-conforming meshes capabilities. Results demonstrate the accuracy, efficiency, and scalability of the implicit scheme in the presence of large scale disparity. The potential of the AMR approach is demonstrated on an island coalescence problem in the high Lundquist-number regime (≥107) with the successful resolution of plasmoid instabilities and thin current sheets.

An Efficient Numerical Solution Method for Detailed Modelling of Large 5th Generation District Heating and Cooling Networks

An Efficient Numerical Solution Method for Detailed Modelling of Large 5th Generation District Heating and Cooling Networks

Efficient Numerical Solution Method for Large Deformation Analyses of Structures Based on the Updated Lagrangian Formulation

AbstractLarge deformation analyses of structures are of great importance to the evaluation of structural performance under extreme environmental loads, but currently available methods are time-cons...

A General and Efficient Method for Solving Regime-Switching DSGE Models

This paper provides a general representation of endogenous and threshold-based regime switching models and develops an efficient numerical solution method. The regime-switching is triggered endogenously when some variables cross threshold conditions that can themselves be regime-dependent. We illustrate our approach using a RBC model with state-dependent government spending policies. It is shown that regime-switching models involve strong non linearities and discontinuities in the dynamics of the model. However, our numerical solution based on simulation and projection methods with regime-dependent policy rules is accurate, and fast enough, to efficiently take into all these challenging aspects. Several alternative specifications to the model and the method are studied.

Numerical solution of stochastic ordinary differential equations using HAAR wavelet collocation method

In the present investigation, we developed an efficient method for numerical solution of stochastic ordinary differential equations (SODE) using Haar wavelets. First, we are analyzing the properties of SODE and Haar wavelets. Next, in order to find the numerical solution of SODE Haar wavelets coefficient and operational matrix of integration is generated and employed. Convergence and error analysis of the proposed method is presented. To demonstrate the strength and effectiveness of the proposed scheme, we give some examples. The numerical computational is conducted to ensure that the method is accurate. The numerical and graphical results obtained are reported, the scheme proposed for studying and finding the solution of SODE is very realistic and accurate computationally.

Multi-leader-follower potential games

In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.

Study on Control Strategy for Tilt-rotor Aircraft Conversion Procedure

This paper studies the control strategy in tilt-rotor aircraft dynamic conversion procedure. A nonlinear flight dynamics model with full flight modes is established. On this basis, a nonlinear optimal control model of dynamic conversion is built by constructing the Bolza form of nonlinear optimal control problem. It contains the limitations and effects of conversion corridor, pilot control, flight attitude, engine rated power, wing stall, and the cooperation between lift and thrust on the procedure of dynamic conversion. An efficient numerical solution method with good convergence is designed to obtain the trajectory and control strategy of dynamic conversion between the modes of helicopter and fixed-wing aircraft. Through the weight parameter analysis, it can be seen that when the subitem of pilot workload is included in the cost function, the displacements of collective stick input as well as the longitudinal stick input are significantly reduced, and the height and pitch attitude change more gently, but the time has been extended. In addition, the pilot workload weight coefficient should not be dominant, otherwise the overly smooth manipulations of collective and longitudinal stick will cause a decrease of height.

Frame decompositions of bounded linear operators in Hilbert spaces with applications in tomography

We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.

Transformed orthogonal functions for solving infinite horizon fractional optimal control problems

In the present paper, an efficient numerical method for solution of infinite horizon fractional optimal control problems is investigated. The fractional derivative in such problems is considered in the Caputo sense. The methodology developed here, is based on utilizing transformed orthogonal functions to approximate the state and control functions directly. Thereby, the original problem is converted to a nonlinear programming problem which can be solved by the well-developed parameter optimization algorithms. While the present solution procedure provides good results and high rate of convergence is achieved, the infinite horizon fractional optimal control problem is solved on the original time interval of the problem, without transforming it to a finite one. Illustrative examples are included at the end and the effectiveness of the proposed methodology is demonstrated.

Thermodynamically compatible model for wavefields simulation in deformed porous medium saturated by a compressible viscous fluid

A computational model for the small amplitude wave propagation in an elastic porous medium saturated by the viscous compressible fluid is discussed. The presented model is an extension of the model [1] and its derivation is based on the symmetric hyperbolic thermodynamically compatible system for two-phase solid-fluid mixture with finite deformations of the solid phase. In the present consideration, the fluid viscosity is taken into account via the unified hyperbolic model for viscous flows with shear stress relaxation. The governing equations form a hyperbolic system written in terms of the mixture velocity, relative velocity of phase motion, pressure and shear stress of the mixture that allows to apply an efficient finite difference method for numerical solution. Some numerical examples are presented, showing physically correct results, and, in particular, the frequency dependence of the shear wave velocity.

Efficient solution techniques for two-phase flow in heterogeneous porous media using exact Jacobians

Two efficient and scalable numerical solution methods will be compared using exact Jacobians to solve the fully coupled Newton systems arising during fully implicit discretization of the equations for two-phase flow in porous media. These methods use algebraic multigrid (AMG) to solve the linear systems in every Newton step. The algebraic multigrid methods rely on (i) a Schur Complement Reduction (SCR-AMG) and (ii) a Constrained Pressure Residual method (CPR-AMG) to decouple elliptic and hyperbolic contributions. Both methods employ automatic differentiation (AD) to calculate exact Jacobians and are compared with finite difference (FD) approximations of the Jacobian. The superiority of AD is shown by several numerical test cases from the field of CO2 geo-sequestration comprising two- and three-dimensional examples. A weak scaling test on JUQUEEN, a BlueGene/Q supercomputer, demonstrates the efficiency and scalability of both methods. To achieve maximum comparability and reproducibility, the Portable Extensible Toolkit for Scientific Computation (PETSc) is used as framework for the comparison of all solvers.

A robust pseudospectral method for numerical solution of nonlinear optimal control problems

In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.

A new operational matrix of Muntz-Legendre polynomials and Petrov-Galerkin method for solving fractional Volterra-Fredholm integro-differential equations

This manuscript is devoted to present an efficient numerical method for finding numerical solution of Volterra-Fredholm integro-differential equations of fractional-order. The technique is based on the M{u}ntz-Legendre polynomials and Petrov-Galerkin method. A new Riemann-Liouville operational matrix for M{u}ntz-Legendre polynomials is proposed using Laplace transform. Employing this operational matrix and Petrov-Galerkin method, the problem transforms to a system of algebraic equations. Next, we solve this system by applying any iterative method. An estimation of the error is proposed. The efficiency and accuracy of the present scheme is illustrated using several examples.

A General and Efficient Method for Solving Regime-Switching DSGE Models

This paper provides a general representation of endogenous and threshold-based regime switching models and develops an efficient numerical solution method. The regime-switching is triggered endogenously when some variables cross threshold conditions that can themselves be regime-dependent. We illustrate our approach using a RBC model with state-dependent government spending policies. It is shown that regime-switching models involve strong non linearities and discontinuities in the dynamics of the model. However, our numerical solution based on simulation and projection methods with regime-dependent policy rules is accurate, and fast enough, to efficiently take into all these challenging aspects. Several alternative specifications to the model and the method are studied.

A fast algorithm for radiative transport in isotropic media

Constructing efficient numerical solution methods for the equation of radiative transfer (ERT) remains as a challenging task in scientific computing despite of the tremendous development on the subject in recent years. We present in this work a simple fast computational algorithm for solving the ERT in isotropic media. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly-averaged ERT solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the ERT solution. The algorithm does not require the underlying medium be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media.