Benchmark Test Examples Research Articles

Retrieving the time-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

PurposeIn order to include the non-negligible lag relaxation time feature that is characteristic of heat transfer in biological bodies, the classical Fourier's law of heat conduction has to be generalized as the Maxwell–Cattaneo law resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the unknown time-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from (non-intrusive) measurements of the temperature on an accessible sub-portion of the boundary that may be taken with an infrared scanner.Design/methodology/approachThe nonlinear and ill-posed problem is reformulated as a nonlinear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite-difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Numerical results for a benchmark test example are presented and thoroughly discussed, shedding light on the performance and effectiveness of the proposed methodology.FindingsThe inverse problem of obtaining the time-dependent blood perfusion coefficient and the temperature in the thermal-wave model of bio-heat transfer from extra boundary temperature measurement has been solved. In particular, the uniqueness of the solution to this inverse problem has been established. Furthermore, our proposed computational method demonstrated successful attainment of the perfusion coefficient and temperature, even when dealing with noisy data.Originality/valueThe originalities of the present paper are to account for such a more representative thermal-wave model of heat transfer in biological bodies and to investigate the possibility of determining its time-dependent blood perfusion coefficient from non-intrusive boundary temperature measurements.

An inverse problem of reconstructing the unknown coefficient in a third order time fractional pseudoparabolic equation

In this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the potential term. The proposed problem is discretized using the cubic B-spline (CB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine $lsqnonlin$ tool has been used to expedite the numerical computations. Both perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

An inverse problem for pseudoparabolic equation: existence, uniqueness, stability, and numerical analysis

In this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

An inverse source problem for a pseudoparabolic equation with memory

<abstract><p>This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine <italic>lsqnonlin</italic>. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.</p></abstract>

Simultaneous identification of the solely time‐dependent potential and source control terms from known moisture moments

Pseudo‐parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo‐parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time‐varying potential and source control terms, in a linear pseudo‐parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least‐squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank–Nicolson finite‐difference method to solve the direct problem. To solve the nonlinear least‐squares minimization problem, we utilize the built‐in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology.

An inverse problem of recovering the heat source coefficient in a fourth-order time-fractional pseudo-parabolic equation

In this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine lsqnonlin tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.

A diffuse interface–lattice Boltzmann model for conjugate heat transfer with imperfect interface

A diffuse interface-lattice Boltzmann model is proposed for modeling conjugate heat transfer problems with imperfect interfaces. According to the continuity condition of heat flux and jump condition of temperature across the imperfect interface, a unified sharp interface equation of the temperature field with a singular source term in the composite media domain is established. Then a diffuse interface equation is obtained by making a smoothed approximation of the unified sharp interface equation. The novel diffuse interface equation is restructured into a second-order heat diffusion equation with an anisotropic diffusion coefficient tensor. To numerically solve the second-order anisotropic diffusion equation, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) scheme is employed. Several numerical simulations of benchmark test examples with analytical solutions are carried out to demonstrate the accuracy of the present model, including the steady-state heat diffusion in a two-layer medium with a flat interface, steady heat diffusion in a square domain with a curved interface and unsteady-state temperature diffusion with two-layer media. The numerical results show that the phenomenon of temperature jump across the imperfect interface can be successfully and correctly captured. Moreover, as an example of practical application, the new proposed diffuse interface model is utilized to predict the effective thermal conductivity of porous media with thermal contact resistance at pore scale. The variation law of the effective thermal conductivity is obtained for each parameter by studying the effects of thermal conductivity ratios of the dual-component material, the shape and volume fraction of blocks in the material, the interfacial thermal conductance coefficient, and the number of blocks on the effective thermal conductivity. Both uniform and random pore structures are considered. The previous theoretical equations strongly support the validity of the present model.

Two-phase inverse Stefan problems solved by heat polynomials method

In this paper, two-phase inverse Stefan problem are investigated using the method of heat polynomials. The approximate solution of the problem is presented in the form of linear combination of heat polynomials. We present numerical results illustrating an application of the heat polynomials method for typical benchmark test examples. Due to ill-posedness of the problem, the regularization will be taken into account. Numerical results are discussed and compared with results obtained by another method.

On an inverse problem for a nonlinear third order in time partial differential equation

In this article, first we convert an inverse problem of determining the unknown timewise terms of nonlinear third order in time partial differential equation (PDE) from knowledge of two boundary measurements to the auxiliary system of integral equations. Then, existence and uniqueness of the solution of this system is proved by means of the contraction principle on a small time interval. Also uniqueness of the solution of the inverse problem is given. However, since the governing equation is yet ill-posed (very slight errors in the temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order in time PDE problem is discretized using the FDM and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. The proposed numerical approach has also been discussed.

An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order

In the present work, we consider an inverse boundary value problem for a two-dimensional pseudo-parabolic equation of the third-order. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. This inverse problem is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis. In addition, the two-dimensional pseudo-parabolic problem is discretized using the FDM and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for benchmark test example is reported and discussed.

Space-dependent heat source determination problem with nonlocal periodic boundary conditions

The purpose of this paper is to identify the space-dependent heat source coefficient numerically, for the first time, in the third-order pseudo-parabolic equation with initial and nonlocal periodic boundary conditions from nonlocal integral observation. This problem emerges significantly in the modelling of numerous phenomena in physics, engineering, mechanics and science. Although, the inverse source problem considered in this article is ill-posed by being sensitive to noise but has a unique solution. For the numerical realization, we apply the finite difference method (FDM) for discretizing the forward problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB subroutine lsqnonlin. Numerical results presented for two benchmark test examples with linear and nonlinear source functions show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the von Neumann stability analysis is also discussed.

Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation

The aim of this paper is to find the time-dependent heat source term numerically in the third-order pseudo-parabolic equation with initial and boundary conditions supplemented by nonlocal integral observation. This is a very interesting and challenging nonlinear inverse source problem with important applications in various fields of mechanics and physics. Unique solvability theorem of this inverse problem is supplied. However, since the problem is still ill-posed (small errors in the integral input data cause large errors in the output force) the solution needs to be regularized. Therefore, to obtain a stable solution, a regularized objective function is minimized to retrieve the unknown heat source coefficient. The third-order pseudo-parabolic problem is discretized using the finite difference method (FDM) and recast as nonlinear least-squares minimization of the Tikhonov regularization function with simple bounds on the unknown force coefficient. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both analytical and perturbed (noisy) data are inverted. Numerical results for three benchmark test examples are presented and discussed. In addition, the von Neumann stability analysis is also discussed.

Long-Term Joint Operation of Cascade Reservoirs Using Enhanced Progressive Optimality Algorithm and Dynamic Programming Hybrid Approach

Dynamic Programming (DP) is one of the most classical methods adopted for reservoir operation. It reduces the computational efforts of complex high-dimensional problems by piecewise dimensionality reduction and provides the global optimums of the problems, but it suffers the “curse of dimensionality”. Progressive Optimality Algorithm (POA) has been used repeatedly in reservoir operation studies during last decades because it alleviates the “curse of dimensionality” of DP and has good convergence and extensive applicability. Nonetheless, the POA encounters two difficulties in multireservoir operation applications. One is the transfer interrupt problem that makes the search procedure hard to achieve free allocation of water between two non-adjacent stages, and another is the dimensionality problem that leads to a low convergence rate. In order to overcome these deficiencies, this study made some enhancements to the POA and proposed a hybrid approach combining the Enhanced POA and DP (EPOA-DP) for long-term operation of cascade reservoir systems. In EPOA-DP, the EPOA is employed to improve the quality of the solutions and DP is used to reduce the computational effort of the two-stage problem solution. The proposed approach was tested using a real-world four-reservoir cascade system and a ten-reservoir benchmark test example. The results demonstrated that it outperforms the POA both in computational time and quality of the solution.

An enhanced binary dragonfly algorithm based on a V-shaped transfer function for optimization of pump scheduling program in water supply systems (case study of Iran)

With the continual growth of population and shortage of energy resources, the optimal consumption of these resources is of particular importance. One of these energy sources is electricity, with a significant amount being used in pumping stations for water distribution systems (WDS). Determining the proper pumping schedule can make significant savings in energy consumption and particularly in costs. This study aims to present an improved population-based nature-inspired optimization algorithmfor pumping scheduling program in WDS. To address this issue, the binary dragonfly algorithm based on a new transfer-function coupled with the EPANET hydraulic simulation model is developed to reduce the energy consumption of pumping stations. The proposed model was firstly implemented and evaluated on a benchmark test example, then on a real water pumping station. Comparison of the proposed method and the genetic algorithm (GA), evolutionary algorithm (EA), ant colony optimization (ACO), artificial bee colony (ABC), particle swarm optimization (PSO), and firefly (FF) was conducted on the benchmark test example, while the obtained results indicate that the proposed framework is more computationally efficient and reliable. The results of the real case study show that while considering all different constraints of the problem, the proposed model can decrease the cost of energy up to 27% in comparison with the current state of operation.

Particle Swarm Optimization and Tabu Search Hybrid Algorithm for Flexible Job Shop Scheduling Problem – Analysis of Test Results

Abstract The paper presents a hybrid metaheuristic algorithm, including a Particle Swarm Optimization (PSO) procedure and elements of Tabu Search (TS) metaheuristic. The novel algorithm is designed to solve Flexible Job Shop Scheduling Problems (FJSSP). Twelve benchmark test examples from different reference sources are experimentaly tested to demonstrate the performance of the algorithm. The obtained mean error for the deviation from optimality is 0.044%. The obtained test results are compared to the results in the reference sources and to the results by a genetic algorithm. The comparison illustrates the good performance of the proposed algorithm. Investigations on the base of test examples with a larger dimension will be carried out with the aim of further improvement of the algorithm and the quality of the test results.

Enhanced accuracy in reduced order modeling for linear stable\unstable system

This paper provides a quantitative measure criterion for selection of poles from a higher order model. The selection of poles is important because it determines both the transient and steady-state information of the dynamical system. The new indices specify which poles are dominant even when they are not the slowest. On the basis of important poles contribution to the system poles, they are selected to form cluster centre. Factor division algorithm is applied to compute the coefficient of the numerator polynomial for determining denominator polynomial. This method extended for the reduction of the unstable system via additive decomposition technique. The proposed method not only can cause less dynamic errors, but also yields zero steady-state error. The efficiency and accuracy of the proposed method are demonstrated through benchmark test examples taken from the literature.

ANT INTELLIGENT APPLIED TO SEWER NETWORK DESIGN OPTIMIZATION PROBLEM: USING FOUR DIFFERENT ALGORITHMS

In this paper, four different kinds of the Ant Colony Optimization Algorithm (ACOA) are used to find optimal solution for sewer network design optimization problem proposing two different formulations for each of them. In both proposed formulations, the decision variables of the problem are cover depths of sewer network nodes. In the second formulation, the constrained version of ACOA is used to find optimal cover depths of the sewer network nodes. The constrained version of ACOA is used here to satisfy slope constraints explicitly leading to reduction of search space of the problem. The Ant System, Elitist Ant System, Elitist-Rank Ant System and Max-Min Ant System are used here to solve two benchmark test examples and the results are presented and compared with other available results. The results show the superiority of the Max-Min Ant System over than other ACOAs in which the trade-off between the two contradictory search characteristic of exploration and exploitation is managed better using this algorithm. Furthermore, best results are obtained with second proposed formulation in comparison with other available results in which they are shown the capability of the proposed formulation to find optimal solution for sewer network design optimization. In other words, the second formulation of Max-Min Ant System has been able to produce results 0.3% and 0.15% cheaper than those obtained by first formulation of Max-Min Ant System for the first and second benchmark test examples, respectively. Furthermore, the average solution cost value of second formulation of Max-Min Ant System is reduced 10.6% (8.1%), 0.43% (0.6%) and 0.34% (0.02%) in comparison with second formulation of Ant System, Elitist Ant System and Elitist-Rank Ant System for first (second) benchmark test examples, respectively.

Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.

Simulation-optimization Model for Design of Water Distribution System using Ant Based Algorithms

In this paper, the Ant Colony Optimization Algorithm (ACOA) is applied to solve Water Distribution System design optimization problem proposing two different methods. Considering pipe diameters as decision variables of the problem, Ant System and Max-Min Ant System, referred to ACOA1 and ACOA2 respectively, are applied to determine pipe diameters. In proposed methods, the ant-based models are interfaced with EPANET as simulator for the hydraulic analysis. Three benchmark test examples are solved with proposed methods and the results are presented and compared with those obtained with other existing methods. The results show the capability of the proposed methods to optimally solve the design optimization problem in which best results are obtained with ACOA2 in comparison with other available results. Furthermore, the results show the superiority of the proposed ACOA2 over than the ACOA1 in which the trade-off between the two contradictory search characteristic of exploration and exploitation is managed better by using Max-Min Ant System.

Different hydraulic analysis conditions for sewer network design optimisation problem using three different evolutionary algorithms

In this paper, the efficiency of considering the constant and varying Manning coefficient for a hydraulic analysis model on the optimal solution of sewer network design optimisation problem is studied. To solve sewer network design optimisation problem, here, different formulations are proposed using genetic algorithm, discreet and continues ant colony optimisation algorithms. In all proposed formulations, the nodal cover depths of the sewer network are taken as decision variables of the problem. Furthermore, for both ant-based algorithms two different formulations are proposed using unconstrained and constrained versions of these algorithms. The constrained versions of these algorithms are proposed here for the explicit satisfaction of the minimum pipe slope constraint leading to smaller search space. Two benchmark test examples are solved here using proposed formulations and the results are presented and compared with other available results. Comparison of the results shows the superiority of considering varying Manning coefficient condition for hydraulic analysis model. Furthermore, the results show the superiority of continues ant colony optimisation algorithm and especially the constrained version of it to optimally solve the sewer network design optimisation problem.