Complex-variable-differentiation Method Research Articles

A novel method for simultaneous determination of thermophysical properties and boundary conditions of phase change problems based on element differential method

Thermophysical properties and boundary conditions for phase change thermal management systems are challenging to be accurately determined, due to the phase change heat transfer phenomenon and complex working conditions. In this work, the element differential method (EDM) and a gradient-based method are combined to simultaneously predict thermal conductivity, mass specific heat, and boundary heat flux in two-dimensional (2D) and three-dimensional (3D) inverse phase change problems, for the first time. The multi-parameter identification for the 3D physical model with phase change is more general than the previous attempts. Moreover, the effective heat capacity method is employed to deal with phase change problems, to improve efficiency. The sensitivity coefficient is accurately determined by the complex-variable-differentiation method (CVDM) in the multi-parameter prediction. Finally, the effect of measurement points, measurement errors, and initial guessed values on the multi-parameter identification are investigated. This study demonstrates that the present method has good accuracy, efficiency, stability, and robustness in dealing with transient nonlinear inverse problems during phase change process.

A modified optimization algorithm for back analysis of properties for coupled stress-seepage field problems

The study is aimed at developing a new algorithm for back analysis of rock mass parameters based on the observed displacements of tunnel excavations under coupled stress-seepage field problems which are generally encountered in hydraulic tunnel projects. The back-analysis algorithm is developed by incorporating the Levenberg-Marquardt optimization technique with complex-variable differentiation method. Additional auxiliary technique is also incorporated to enhance the convergence and stability of the proposed algorithm during the back-estimation of multiple rock mass parameters. A hypothetical hydraulic tunnel case was used for testing and validating the proposed algorithm by incorporating the method in a finite element code in which multiple rock mass parameters (such as modulus, permeability, and in situ stress) were treated as target parameters. Results show that the multiple rock mass parameters can be accurately and efficiently estimated by back analysis using a newly developed algorithm for coupled stress and seepage fields encountered in tunneling. The proposed algorithm can be used for predicting excavation behavior, particularly, the stress-induced deformations at subsequent stages of tunnel excavation under coupled multiple fields (e.g. stress and seepage fields).

A general method for predicting the bank thickness of a smelting furnace with phase change

A new methodology is proposed for predicting the bank thickness covering and protecting the refractory brick walls of the smelting furnaces. The inverse method predicts the bank thickness changing with both time and coordinates, by using a two-dimensional physical model of the furnace that is more general and challenging than the previous studies. Moreover, the thermal conductivities of the phase change material are identified simultaneously because they are poorly known, which is more challenging and practical. The inverse problem is solved by the Levenberg-Marquardt Method (LMM). The direct problem of the smelting furnace and the sensitivity matrix coefficients of the LMM are calculated by ABAQUS, using a complex user-defined element that is set up based on the complex-variable-differentiation method (CVDM) and the user element subroutine (UEL). Finally, numerical examples are given to examine the performances of the approach for predicting the bank thickness covering the refractory brick walls of the smelting furnaces.

A general approach for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures

A novel general approach is proposed for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures. The complex-variable-differentiation method (CVDM) is introduced into the commercial finite element method (FEM) software ABAQUS, for solving three-dimensional inverse heat conduction problems for the first time. First, the complex variable finite elements are set up, and the complex variable FEM is innovatively developed through the user element subroutine (UEL) in ABAQUS. Then, the key parameters in the Levenberg-Marquardt algorithm for solving inverse problems – sensitivity matrix coefficients – are accurately evaluated by utilizing the CVDM and ABAQUS standard solvers with the UEL. Finally, several numerical tests are carried out to verify the performances of the new approach for solving the three-dimensional transient nonlinear inverse heat conduction problem in an irregular complex structure.

Simultaneous estimation of hydro-dipersive parameters using a new modified levenberg-marquardt algorithm

Determination of soil hydrodynamic parameters at field scale is of great importance for modeling soil water dynamics and for agricultural water management. The direct estimation of those parameters is time-consuming and afflicted with serious uncertainties. Inverse modeling is known to get efficient technique for solving non-linear problems in hydrology. Levenberg–Marquardt (LM) algorithm is a gradient-based method, which has been widely used for solving inverse soil water flow problems. In the LM algorithm, sensitivity coefficients are mainly evaluated by numerical differentiation methods. However, sensitivity coefficients are difficult to be precisely calculated by numerical differentiation methods, if transient states and non-linearities are involved. In this paper, a new approach is proposed for sensitivity analysis using the complex variabledifferentiation method (CVDM) to estimate simultaneously the hydraulic and dispersive properties of unsaturated soil from in-situ experiments. In this approach, the sensitivity coefficients can be determined in a more accurate way than the traditional finite difference method. The results show that the new inverse analysis method in the present work has high accuracy, validity, uniqueness and higher inversion efficiency, compared with the previous least-squares method. The simulated and measured water contents and tracer concentration were generally close. Overall, it was concluded that the CVDM is a promising method to estimate hydro-dispersive parameters in the unsaturated zone.

A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.

Efficient hypersingular line and surface integrals direct evaluation by complex variable differentiation method

We present an efficient numerical scheme to evaluate hypersingular integrals appeared in boundary element methods. The hypersingular integrals are first separated into regular and singular parts, in which the singular integrals are defined as limits around the singularity and their values are determined analytically by taking the finite part values. The remaining regular integrals can be evaluated using rational interpolatory quadrature or complex variable differentiation (CVDM) for the regular function when machine precision like accuracy is required. The proposed method is then generalised for evaluating hypersingular surface integrals, in which the inner integral is treated as the hypersingular line integral via coordinate transformations. The procedure is implemented into 8-node rectangular boundary element and 6-node triangular element for numerical evaluation. Finally, several numerical examples are presented to demonstrate the efficiency of the present method. To the best of our knowledge, the proposed method is more accurate, faster and more generalised than other methods available in the literature to evaluate hypersingular integrals.

High Order Projection Plane Method for Evaluation of Supersingular Curved Boundary Integrals in BEM

Boundary element method (BEM) is a very promising approach for solving various engineering problems, in which accurate evaluation of boundary integrals is required. In the present work, the direct method for evaluating singular curved boundary integrals is developed by considering the third-order derivatives in the projection plane method when expanding the geometry quantities at the field point as Taylor series. New analytical formulas are derived for geometry quantities defined on the curved line/plane, and unified expressions are obtained for both two-dimensional and three-dimensional problems. For the two-dimensional boundary integrals, analytical expressions for the third-order derivatives are derived and are employed to verify the complex-variable-differentiation method (CVDM) which is used to evaluate the high order derivatives for three-dimensional problems. A few numerical examples are given to show the effectiveness and the accuracy of the present method.

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.

A Modified Conjugate Gradient Method for Transient Nonlinear Inverse Heat Conduction Problems: A Case Study for Identifying Temperature-Dependent Thermal Conductivities

Despite numerous studies of conjugate gradient methods (CGMs), the “sensitivity problem” and the “adjoint problem” are inevitable for nonlinear inverse heat conduction problems (IHCPs), which are accompanied by some assumptions and complicated differentiating processes. In this paper, a modified CGM (MCGM) is presented for the solution of a specified transient nonlinear IHCP, to recover temperature-dependent thermal conductivities for a case study. By introducing the complex-variable-differentiation method (CVDM) for sensitivity analysis, the sensitivity problem and the adjoint problem are circumvented. Five test examples are given to validate and assess the performance of the MCGM.

Inverse analysis of thermal conductivities in transient non-homogeneous and non-linear heat conductions using BEM based on complex variable differentiation method

This paper presents a new inverse analysis approach to sensitivity analysis and material property identification in transient non-homogeneous and non-linear heat conduction Boundary Element Method (BEM) analysis based on Complex Variable Differentiation Method (CVDM). In this approach, the material properties are taken as the optimization variables, and the sensitivity coefficients are computed by CVDM. The advantages of using CVDM are that the computation of partial derivatives of an implicit function is reduced to function calculation in a complex domain, and the parameter sensitivity coefficients can be determined in a more accurate way than the traditional Finite Difference Method (FDM). Based on BEM and CVDM in evaluation of the sensitivity matrix of heat flux, the parameter such as thermal conductivity can be accurately identified. Six numerical examples are given to demonstrate the potential of the proposed approach. The results indicate that the presented method is efficient for identifying the thermal conductivity with single or multiple parameters.

Development of complex-variable differentiation method and its application in isogeometric analysis

The complex-variable differentiation method (CVDM) is a very promising method to compute derivatives of complicated functions and is free from cancellation errors which often occur in the finite difference method. In this paper, CVDM is developed for the two-variable situation and is first applied to the isogeometric analysis method in which the computation of derivatives of rational functions represented by non-uniform rational B-splines (NURBS) method is very complicated. An example about h-refinement is given and the computational results show that the derived formulations in the paper is correct and the computation of derivatives using CVDM in isogeometric analysis is much simpler than using the traditional NURBS derivative formula.

Application of the Complex Variable Differential Method in the Back Analysis of Geotechnical Engineering

On the stability analysis of complex slope, the determination of rock-soil masses shear strength parameters is very important. It’s very necessary to use the back analysis method to verify experiment result of rock-soil masses parameters for the important slope engineering. Because the sensitivity of shear strength parameters to displacement parameters is very poor, so that we couldn’t get the satisfactory shear strength parameters result by traditional back analysis method. Aiming at the problems of traditional back analysis method, this paper puts forward a totally new method of back analysis, which is applicable to the shear strength parameters of rock-soil masses through the integration of complex variable differentiation method, optimization method and elastic-plasticity finite-element method. The method mathematically back calculates shear strength parameters of rock-soil masses on the basis of displacement of measuring point. The sample calculation result indicates that the method possesses high accuracy and searching efficiency, and is a method of back analysis of displacement deserving popularizing.

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

AbstractIn this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45:463–484; Int. J. Numer. Meth. Fluids 2005; 47:19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45:463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47:19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.

A new inverse analysis approach for multi-region heat conduction BEM using complex-variable-differentiation method

This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). In this approach, the material properties and coordinates of an unknown region boundary are taken as the optimization variables, and the sensitivity coefficients are computed by the Complex-Variable-Differentiation Method (CVDM). Due to the use of CVDM, the sensitivity coefficients can be accurately determined in a way that is as simple to use as the Finite Difference Method (FDM) and an inverse analysis for a complex composite structure can be easily performed through a similar procedure to the direct computation. Although basic integral equations are presented for heat conduction problems, the application of the proposed algorithm to other problems, such as elastic problems, is straightforward. Two numerical examples are given to demonstrate the potential of the proposed approach.