Numerical Inverse Method Research Articles

Measurement of convective heat transfer coefficients in a randomly packed bed of silica gel particles using IHTP analysis

The purpose of this research is to determine the convective heat transfer coefficient (ht) for airflow through two randomly packed beds of microporous silica gel particles of nearly uniform diameter (i.e., for the average diameter of 1.60 and 2.58mm). The experimental test facility was developed to induce a transient step change in the inlet uniform airflow while the temperature was measured at several locations within the silica gel beds (x/L=0.29, 0.48, 0.69). The transient test data were simulated numerically using the volume averaging method for air flow through a homogeneous packed bed of particles for a range of Reynolds numbers (40<Redh<100). The convective heat transfer coefficient was determined by using the numerical inverse method to minimize the difference between the simulated and experimental temperature profiles. The sensitivity analysis stated that the thermocouple placed at the middle of the test section (x/L=0.48) had the highest sensitivity during the first 400s of transient testing. Using the ht values obtained by the inverse calculations, a good agreement is obtained within the experimental uncertainty limits for the experimental and numerical temperature temporal profiles at different locations within the beds; but the agreement is best using the transient temperature data at x/L=0.48. Based on the ht values, a new correlation between Nusselt number and Reynolds number is presented with lower uncertainty (the maximum uncertainty of ±30%) compared to the correlations in the literatures. It was found that the proposed correlation is in agreement with the data in literature within their uncertainty bounds.

Modelling, detection and identification of flexural crack damages in beams using frequency response functions

In this paper, the original receptance method first proposed by Bishop (The mechanics of vibration. Cambridge University Press, Cambridge, 1979) is further generalized and extended to the vibration modelling of stepped beams and truss structures with multiple crack damages. Cracks are physically treated as rotational springs representing local flexibilities at crack locations induced by crack depths. Frequency response functions (FRFs) with any given number of crack locations and depths are first established from which numerically exact natural frequencies and mode shapes are extracted. Since these FRFs can be measured in practice in a typical modal testing, a crack location method is then developed by using rotational FRFs which can be estimated from these measured translational FRF data. Substantial slope discontinuities are observed and are further employed to construct a crack location matrix to locate cracks. Finally, to identify crack parameters such as crack depths, a new Numerical Inverse FRF Sensitivity method is developed. Unlike the case of analytical model updating where an analytical model is available from which sensitivities can be derived, FRF sensitivities needed in this case have to be numerically computed. Such numerical sensitivity approach is believed to have wider applications in inverse dynamics problems such as analytical model updating. A stepped beam and a GARTEUR structure are employed in numerical case studies and very encouraging results have been achieved even when FRF data are contaminated with substantial measurement noise.

Failure Performance Analysis of Resistance Spot Welded Advanced High Strength Steel Sheets

A numerical method to predict failure performance was developed for resistance spot welded AHSS (advanced high strength steel) sheets in this work and applied for a TRIP (transformation induced steel sheets) 980 sheet. For the numerical analysis, utilizing the numerical inverse calibration method, hardening data and fracture criteria were characterized for the base and the weld nugget based on the standard simple tension test and a newly designed miniature tension test, respectively. The characterized properties of the base and the weld nugget were then applied to analyze failure performance in the coupon tests of the lap-shear and U-shape tension tests carried out for welded sheets. The analysis was performed under the quasi-static condition in this early effort and showed reasonably good agreement with experiments both in failure modes and strength.

A novel inverse method for identification of 3D thermal conductivity coefficients of anisotropic media by the boundary element analysis

This paper presents a new numerical inverse method for identifying the six components of the thermal conductivity tensor of a 3D anisotropic medium with an arbitrary shape. The unknowns are inversely calculated using heat conduction problems with extra information at some boundary sampling points. For better stability, the inverse method uses data supplied from more than one steady-state heat conduction problem. Since all sampling points are taken to be on boundary surfaces for the sake of easy access, the boundary element method (BEM) for direct calculation is employed for the sensitivity analyses. The off-diagonal components of the thermal conductivity tensor must satisfy three nonlinear inequality constraints, which make the inverse analysis even more challenging. To overcome this difficulty, the inverse problem is formulated in terms of the principal thermal conductivities along with the rotation angles of principal axes, by which the three constraints disappear. For the inverse analysis, the damped Gauss–Newton method is adopted for the optimization process. In the end, numerical examples are presented, showing that the proposed method can yield reliable solutions even in cases with relatively large measurement error and with initial guesses far from the exact solution.

High-throughput determination of the composition-dependent interdiffusivities in Cu-rich fcc Cu–Ag–Sn alloys at 1073 K

Based on the recently developed pragmatic numerical inverse method for determining the composition-dependent interdiffusivities in a ternary system by using a single diffusion couple, high-throughput determination of the interdiffusivities in Cu-rich fcc Cu–Ag–Sn alloys at 1073K was performed in the present work. The composition-dependent interdiffusivity matrices along the entire diffusion paths of five fcc Cu–Ag–Sn diffusion couples were obtained. The reliability of the interdiffusivities determined by the pragmatic numerical inverse method was first validated by Fick’s second law applied to numerical simulation of composition profiles and interdiffusion fluxes for each diffusion couple. The excellent agreement between the simulated results and the experimental data was obtained. In order to further validate their reliability, the traditional Matano–Kirkaldy method was employed to evaluate the interdiffusivities at the intersection point of the diffusion paths for every two diffusion couples. The good agreement between the interdiffusivities determined by the pragmatic numerical inverse method and those by the traditional Matano–Kirkaldy method was also observed. These facts indicate that the interdiffusivities determined by the pragmatic numerical inverse method are reliable.

A novel approach to forced vibration behavior of thick-walled cylinders

This study is to investigate the effect of anisotropy on the forced vibration behavior of hollow cylinders under dynamic internal pressure. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin's numerical inversion method. Durbin's numerical inverse method into the analysis of transient thermal stresses in annular fins is a novel approach. Durbin's numerical inverse method successfully implements the boundary value problem which can be solved in Laplace space. Various material models from the literature are used and corresponding radial displacement distributions and stress distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases and virtually exact results are obtained. The anisotropy constant is a useful parameter from a design point of view in that it can be tailored for specific applications to control the stress distribution.

Novel Approach to Transient Thermal Stress in an Annular Fin

This study is to investigate the transient thermal stresses of an annular fin with its base subjected to a heat flux of a decayed exponential function of time. The problems are solved analytically in the Laplace domain, and the results obtained are transformed to the real-time space using the modified Durbin’s numerical inversion method. Durbin’s numerical inverse method into the analysis of transient thermal stresses in annular fins is a novel approach. Durbin’s numerical inverse method successfully implements the boundary value problem, which can be solved in Laplace space. Various material models from the literature are used, and corresponding temperature distributions and stress distributions are computed. Verification of the proposed method is done using benchmark solutions available in the literature for some special cases, and virtually exact results are obtained.

Tube hydroforming compression test for friction estimation—numerical inverse method, application, and analysis

Friction plays an important role in forming processes, in fact it influences the material flow and therefore it affects the process and part characteristics. In particular, friction is a very influencing factor in tube hydroforming (THF), where high die–part contact pressure and area make the material sliding very difficult. As a consequence, the material hardly flows to the expansion zones and the part formability can be compromised. To obtain sound parts, FEM models allow the study of the process and optimize its parameters, but they require the right definition of the friction at tube–die interface. For these reasons, friction represents a key-point in THF processes and its knowledge and prediction are very important even if, nowadays, a comprehensive friction test for THF is not available in literature. With this paper, the authors want to propose and evaluate a method to estimate friction for THF processes. In particular, a numerical inverse method allowing the estimation of the Coulombian friction coefficient combining experimental test and FE simulation results will be described. The method is based on the effects of friction on the tube final thickness distribution when it is pressurized and compressed by two punches under different lubrication conditions without expansion. In particular, how the use of few and fast FE simulations allows to estimate an analytical function that takes into account the process conditions and that can be used in combination with experimental results in order to estimate the friction coefficient in THF processes will be shown.

An optimisation scheme based on the local interaction simulation approach and Lamb waves for elastic property estimation in multi-layered composites

The paper demonstrates the application of guided ultrasonic waves for estimating material elastic properties in mul- ti-layered composite plates. Two-dimensional Lamb wave propagation is modelled using the Local Interaction Simulation Approach (LISA). The wave responses are then used to estimate relevant material properties. This numerical inverse method is based on an optimisation scheme using a Differential Evolution (DE) algorithm. The method is demonstrated on a simulated seven-layered glass fibre-epoxy composite plate. The results show a good agreement between the exact and reconstructed ma- terial properties.

Parameters identification in strain-rate and thermal sensitive visco-plastic material model for an alumina dispersion strengthened copper

The main objective of this paper is getting strain-hardening, thermal and strain-rate parameters for a material model in order to correctly reproduce the deformation process that occurs in high strain-rate scenario, in which the material reaches also high levels of plastic deformation and temperature. In particular, in this work the numerical inverse method is applied to extract material strength parameters from experimental data obtained via mechanical tests at different strain-rates (from quasi-static loading to high strain-rate) and temperatures (between 20 °C and 1000 °C) for an alumina dispersion strengthened copper material, which commercial name is GLIDCOP ®. Thanks to its properties GLIDCOP ® finds several applications in particle accelerator technologies, where problems of thermal management, combined with structural requirements, play a key role. Currently, it is used for the construction of structural and functional parts of the particle beam collimation system. Since the extreme condition in which the material could operate, it is fundamental to characterize it in a wide range both in strain-rate and temperature. The numerical inverse method used in this work is particularly useful to reproduce experimental results when the stress–strain fields in the specimen cannot be correctly described via analytical models. Furthermore this procedure is useful to take into account thermal phenomena generally affecting high strain-rate tests in which the heat conversion of plastic work produces an adiabatic overheating. So, the applicability of this method is particularly indicated in special fields, such as aerospace engineering, ballistic, crashworthiness studies or particle accelerator technologies. The attention is focused on evaluating the most suitable strategy of material model parameters optimization to obtain the best fit between experimental data and numerical results. In this regards, it is important to determine which material model coefficients can be considered as optimization variables and for each of them the most suitable range of variation.

Numerical Inverse Method for Friction Coefficient Estimation in Tube Hydroforming

Friction plays an important role in forming processes, in fact it influences the material flow and therefore it affects the process and part characteristics. In particular, friction is a very influencing factor in Tube Hydroforming (THF), where high die-part contact pressure and area make the material sliding very difficult. As a consequence, the material hardly flows to the expansion zones and the part formability can be compromised. To obtain sound parts, FEM models allow to study the process and optimize its parameters, but they require the right definition of the friction at tube-die interface. For these reasons, friction represents a key-point in THF processes and its knowledge and prediction are very important even if, nowadays, a comprehensive friction test for THF is not available in literature. With this paper, the Authors want to propose a novel approach to estimate friction for THF processes. In particular it will be described a numerical inverse method that allows to estimate the Coulombian friction coefficient combining experimental test and FE simulation results. The method is based on the effects of friction on the tube final thickness distribution when it is pressurized and compressed by two punches under different lubrication conditions without expansion. In particular, it will be shown how the use of few and fast FE simulations allows to estimate an analytical function that takes into account the process conditions and that can be used in combination with experimental results in order to estimate the friction coefficient in THF processes.

Determination of heat transfer coefficients by extrapolation and numerical inverse methods in squeeze casting of magnesium alloy AM60

In this work, a different wall-thickness 5-step (with thicknesses as 3, 5, 8, 12, 20 mm) casting mold was designed, and squeeze casting of magnesium alloy AM60 was performed under an applied pressure 30, 60 and 90 MPa in a hydraulic press. The casting–die interfacial heat transfer coefficients (IHTC) in the 5-step casting were determined based on thermal histories throughout the die and inside the casting which were recorded by fine type-K thermocouples. With measured temperatures, heat flux and IHTCs were evaluated using the polynomial curve fitting method and numerical inverse method. For numerical inverse method, a solution algorithm was developed based on the function specification method to solve the inverse heat conduction equations. The IHTCs curves for five steps versus time were displayed. As the applied pressures increased, the IHTC peak value of each step was increased accordingly. It can be observed that the peak IHTC value decreased as the step became thinner. Furthermore, the accuracy of these curves was analyzed by the direct modeling calculation. The results indicated that heat flux and IHTCs determined by the inverse method were more accurately than those from the extrapolated fitting method.

Two-level time-domain decomposition based distributed method for numerical solutions of pharmacokinetic models

In order to predict variations of drug concentration during a given period of time, numerical solutions of pharmacokinetic models need to be obtained efficiently. Analytical solutions of linear pharmacokinetic models are usually obtained using the Laplace transform and inverse Laplace tables. The derivations of solutions to complex nonlinear models are tedious, and such solution process may be difficult to implement as a robust software code. For nonlinear models, the fourth-order Runge–Kutta ( RK4) is the most classical numerical method in obtaining approximate numerical solutions, which is impossible to be implemented in distributed computing environments without much modification. The reason is that numerical solutions obtained by using RK4 can only be computed in sequential time steps. In this paper, time-domain decomposition methods are adapted for nonlinear pharmacokinetic models. The numerical Inverse Laplace method for PharmacoKinetic models ( ILPK) is implemented to solve pharmacokinetic models with iterative inverse Laplace transform in each time interval. The distributed ILPK algorithm, which is based on a two-level time-domain decomposition concept, is proposed to improve its efficiency. Solutions on the coarser temporal mesh at the top level are obtained one by one, and then those on the finer temporal mesh at the bottom level are calculated concurrently by using those initial solutions that have been obtained at the top level decomposition. Accuracy and efficiency of the proposed algorithm and its distributed equivalent are investigated by using several test models. Results indicate that the ILPK algorithm and its distributed equivalent are good candidates for both linear and nonlinear pharmacokinetic models.

Improved numerical inverse kinematics for human pose estimation

We propose a real-time pose estimation method that addresses the weaknesses of the numerical inverse kinematics method. Using conventional inverse kinematics based on the numerical method requires many iterations; moreover, a singularity in the Jacobian matrix as well as a local minimum problem can occur. To solve these problems, we propose an inverse kinematics method combined with an unscented Kalman filter (UKF) to recover intermediate joint information. Because the numerical inverse kinematics method optimizes a state, the solution can often converge to the local minimum and require many iterations. We use several sigma points for analysis to find the optimum state by using an unscented transform. The improved method using a UKF converges faster than the numerical inverse kinematics method for the global minimum of the existing inverse kinematics. We use 2-D image processes to extract body areas from the input images, and a 3-D reconstruction algorithm is used to estimate the 3-D positions of the extracted human body area. Using the improved method, we generate intermediate joints for each body part and the results show that the proposed method reduces the computational complexity and increases the accuracy of estimation compared to conventional numerical inverse kinematics.

Numerical Inverse Heat Transfer Analysis for Temperature-Sensitive-Paint Measurements in Hypersonic Tunnels

This paper describes a numerical inverse method used in temperature-sensitive-paint measurements for quantitative global heat flux diagnostics in hypersonic tunnels. An iterative algorithm is developed to solve the one-dimensional inverse heat transfer problem for unsteady heat flux into a polymer layer (temperature-sensitive paint) on a semi-infinite base, where the temperature dependencies of the thermal properties of materials are taken intoaccount.Surfaceheat fluxdistributionsonconemodelsarecalculatedbyusingthisalgorithmfromtemperaturesensitive-paint images acquired in hypersonic wind tunnel testing. Nomenclature a = thermal diffusivity c = specific heat I = luminescent intensity k = thermal conductivity L = polymer thickness qs = surface heat flux qs� 1D� = surface heat flux obtained by using the onedimensional method T = temperature Tin = initial temperature t = time x, z = coordinates on surface y = coordinate in normal direction to surface

Identification of strain-rate and thermal sensitive material model with an inverse method

This paper describes a numerical inverse method to extract material strength parameters from the experimental data obtained via mechanical tests at different strain- rates and temperatures. It will be shown that this procedure is particularly useful to analyse experimental results when the stress-strain fields in the specimen cannot be correctly described via analytical models. This commonly happens in specimens with no regular shape, in specimens with a regular shape when some instability phenomena occur (for example the necking phenomena in tensile tests that create a strongly heterogeneous stress-strain fields) or in dynamic tests (where the strain-rate field is not constant due to wave propagation phenomena). Furthermore the developed procedure is useful to take into account thermal phenomena generally affecting high strain-rate tests due to the adiabatic overheating related to the conversion of plastic work. The method presented requires strong effort both from experimental and numerical point of view, anyway it allows to precisely identify the parameters of different material models. This could provide great advantages when high reliability of the material behaviour is necessary. Applicability of this method is particularly indicated for special applications in the field of aerospace engineering, ballistic, crashworthiness studies or particle accelerator technologies, where materials could be submitted to strong plastic deformations at high-strain rate in a wide range of temperature. Thermal softening effect has been investigated in a temperature range between 20°C and 1000°C.

Numerical inverse method predicting acoustic spinning modes radiated by a ducted fan from free-field test data

Spinning modes generated by a ducted turbofan at a given frequency determine the acoustic free-field directivity. An inverse method starting from measured directivity patterns is interesting in providing information on the noise sources without requiring tedious spinning-mode experimental analyses. According to a previous article, equations are based on analytical modal splitting inside a cylindrical duct and on a Rayleigh or a Kirchhoff integral on the duct exit cross section to get far-field directivity. Equations are equal in number to free-field measurement locations and the unknowns are the propagating mode amplitudes (there are generally more unknowns than equations). A MATLAB procedure has been implemented by using either the pseudoinverse function or the backslash operator. A constraint comes from the fact that squared modal amplitudes must be positive which involves an iterative least squares fitting. Numerical simulations are discussed along with several examples based on tests performed by Rolls-Royce in the framework of a European project. It is assessed that computation is very fast and it well fits the measured directivities, but the solution depends on the method and is not unique. This means that the initial set of modes should be chosen according to any known physical property of the acoustic sources.

An inverse method for the determination of full field stresses from experimentally measured normal strains

Certain diffraction-based techniques that measure strains in bulk samples are limited to determination of normal strains. A numerical inverse method is developed to determine full field stresses from the experimentally determined normal strains in isotropic solids under plane stress conditions. The method is based on satisfying the equations of equilibrium and the constitutive relations. The finite difference method is employed to solve the equations and to determine the complete stress field. Furthermore, a least-squares procedure is used to determine the unknown functions of integration in conjunction with known values of shear stress along a reference line. The method is verified by using the normal strain fields in various specimens obtained using both finite element analysis and exact elasticity solutions. It is found that the proposed method predicts the shear stresses accurately in the examples considered.

Guided ultrasonic propagation in the tri-layer structures. Application to the reconstruction of their material properties from dispersion curves

This paper deals with the guided ultrasonic propagation in the tri-layer structures. The propagation model and the procedure used to determine the dispersion curves and the displacement field in the structure are those presented earlier in B. Bougaze et al. [Eur. Phys. J. Appl. Phys. 32 , 207 (2005)] and tested for the case of (aluminium/adhesive/aluminium) structure. Here, we consider the case of a tri-layer structure presenting two different adherents: copper and steel. We adjusted our programs for this case and determined the dispersion curves and the displacement field in the structure. The obtained dispersion curves are then used to reconstruct the material properties of the tri-layer structure via a proposed numerical inverse method based on an optimization process. These material properties include the longitudinal and shear wave velocities as well as the density of each layer of the structure. The agreement between the reconstructed and the exact material properties is very well demonstrating the efficiency of the proposed numerical inverse method.

Prediction of chloride ingress into saturated concrete on the basis of a multi-species model by numerical calculations

A multi-species model based on the Nernst-Planck equation has been developed by using a finite volume method. The model makes it possible to simulate transport due to an electrical field or by diffusion and to predict chloride penetration through water saturated concrete. The model is used in this paper to assess and analyse chloride diffusion coefficients and chloride binding isotherms. The experimental assessment of the effective chloride diffusion coefficient consists in measuring the chloride penetration depth by using a colorimetric method. The effective diffusion coefficient determined numerically allows to correctly reproduce the chloride penetration depth measured experimentally. Then, a new approach for the determination of chloride binding, based on non-steady state diffusion tests, is proposed. The binding isotherm is identified by a numerical inverse method from a single experimental total chloride concentration profile obtained at a given exposure time and from Freundlich's formula. In order to determine the initial pore solution composition (required as initial conditions for the model), the method of Taylor that describes the release of alkalis from cement and alkali sorption by the hydration products is used here. Finally, with these input data, prediction of total and water-soluble chloride concentration profiles has been performed. The method is validated by comparing the results of numerical simulations to experimental results obtained on various types of concretes and under different exposure conditions.