Numerical Inverse Method Research Articles

Analysis of the accuracy of the inverse marching method used to determine thermal stresses in cylindrical pressure components

This paper analyses the inverse marching method used to determine the thermal stresses on the inner surface of a thick-walled cylindrical element not weakened by holes in the transient state. The heat conduction problem was considered one-dimensional, i.e. it was assumed that heat is transferred only in the radial direction. The method is based on measuring the temperature inside the pipeline wall at a single point and assuming that the pipeline is thermally insulated. The paper undertook an evaluation of the influence of the measuring point's distance from the inner surface, the number of control volumes into which the inverted area was divided and the length of the time step on the accuracy of the calculated temper-ature, heat transfer coefficient and thermal stresses on the inner surface of the pressure element. Verification was performed by comparing the calculation results obtained from the direct analytical method perturbed by random errors with those obtained from the numerical inverse step method.

High-throughput determination of interdiffusivity and atomic mobilities in bcc Ti–Cr–Mo alloys

A comprehensive understanding of the diffusion coefficients and atomic mobilities in the Ti–Cr–Mo system is crucial for establishing an atomic mobility database for titanium-based high-strength titanium alloys. In the present work, nine sets of solid/solid diffusion couples were prepared and measured in bcc Ti–Cr–Mo alloys, with the temperatures ranging from 1373 to 1473 K. Scanning electron microscopy (SEM), and Electro Probe Micro-Analysis (EPMA) techniques were used to respectively test its microstructure and composition distance. The diffusion coefficients in bcc Ti–Cr–Mo alloys were determined using the Matano-Kirkaldy method and numerical inverse method respectively. The diffusion coefficients obtained from the above two methods exhibit good consistency. The atomic mobility parameters for the Ti–Cr–Mo systems were provided with the assistance of HitDIC software. A three-dimensional plot of interdiffusivities was generated for chromium and molybdenum contents ranging from 0 to 20 mol % and 0–14 mol %. Moreover, the application of the Arrhenius equation enabled the determination of changes in frequency factors and activation energies concerning temperature and composition.

Interdiffusion in BCC_B2 Ni–Ti–V alloys at 1223K–1323K

Establishing a Ni–Ti–V system diffusion kinetics database contributes to a better understanding of the precipitation processes in shape memory alloys. In this paper, eleven sets of BCC_B2 Ni–Ti–V single-phase diffusion couples were prepared and annealed at temperatures of 1223 K, 1273 K, and 1323 K. The composition-distance curves were determined using the electron probe microstructure analysis technique. The interdiffusion coefficients were extracted using the HitDIC software, which is based on the numerical inverse method. The software demonstrated high accuracy in predicting the composition curves of the prepared diffusion couples. The interdiffusion coefficients inferred from the HitDIC software and the ones calculated by the Matano-Kirkaldy method achieved good agreement. This demonstrates the reliability and rationality of the evaluated interdiffusion coefficients. This study also investigated the composition and temperature dependencies of interdiffusion coefficients.

High-throughput determination of the interdiffusion coefficients and atomic mobilities in bcc Ti–Fe–V alloys

A comprehensive understanding of diffusion kinetics of the Ti–Fe–V system is crucial for elucidating the solidification process in high-strength titanium alloys. In this paper, ten sets of diffusion couples within bcc Ti–Fe–V alloys that underwent annealing at temperatures of 1273K, 1323K, and 1373K were prepared. Composition-distance curves were determined using the electron probe microstructure analysis technique. Atomic mobility parameters and interdiffusion coefficients were obtained using the numerical inverse method with the assistance of HitDIC software. It should be noted that the atomic mobility parameters for Ti–Fe and Ti–V were reoptimized using the most up-to-date thermodynamic descriptions. This approach enables the acquisition of comprehensive data regarding the temperature-composition relationship in diffusion. D˜FeFeTi exhibits a markedly greater value compared to D˜VVTi. Both D˜FeFeTi and D˜VVTi demonstrate an increase with the increase of temperature. Furthermore, the Arrhenius equation can be applied to solve for the variation in frequency factors and activation energies concerning temperature and composition.

The impact of fractional derivative on thermomechanical interactions in two-dimensional skin tissues throughout hyperthermia treatment

This work aims to study the influence of fractional derivatives and thermal relaxation time on two-dimensional biological tissues using an eigenvalue-based approach. Understanding the heat transfer mechanism and its thermomechanical interactions with the skin tissue of patients is crucial for the successful implementation of thermal treatment procedure. The study employs Laplace and Fourier transforms, and the resulting formulations are applied to human tissues undergoing regional hyperthermia treatment for tumor therapy. The numerical inversion process for Laplace and Fourier transformations utilizes the Stehfest numerical inverse method. The results demonstrate that thermal relaxation time and fractional order parameter significantly influence all analyzed distributions. The numerical findings suggest that thermo-mechanical waves propagate through the skin tissue over finite distances, which helps mitigate the unrealistic predictions made by the Pennes' model.

Generalized thermomechanical interaction in two-dimensional skin tissue using eigenvalues approach

The aim of this work is to analytically study the thermo-mechanical response of two-dimensional skin tissues when subjected to instantaneous heating. A complete understanding of the heat transfer process and the associated thermal and mechanical effects on the patient's skin tissues is critical to ensuring the effective applications of thermal therapy techniques and procedures. The surface boundary of the half-space undergoes a heat flux characterized by an exponentially decaying pulse, while maintaining a condition of zero traction. The utilization of Laplace and Fourier transformations is employed, and the resulting formulations are then applied to human tissues undergoing regional hyperthermia treatment for cancer therapy. To perform the inversion process for Laplace and Fourier transforms, a numerical programming method based on Stehfest numerical inverse method is employed. The findings demonstrate that blood perfusion rate and thermal relaxation time significantly influence all the analyzed distributions. Numerical findings suggest that thermo-mechanical waves propagate through skin tissue over finite distances, which helps mitigate the unrealistic predictions made by the Pennes' model.

Experimental determination and high-throughput calculation of the interdiffusion coefficient matrix and atomic mobility in Ag-rich fcc Ag–Sn–Zn alloys

In this paper, 10 groups of ternary Ag–Sn–Zn diffusion couples were prepared under 873, 973 and 1073 K within the fcc single-phase region. The element distribution across the diffusional interface was measured by EPMA, and a high-throughput determination of atomic mobilities in Ag-rich fcc Ag–Sn–Zn alloys was performed by using HitDIC (High-Throughput Determination of Interdiffusion Coefficients, https://hitdic.com/) software in the framework of the numerical inverse method. A self-consistent set of atomic mobility parameters was obtained. Reliability of the evaluated atomic mobility parameters were further verified by reasonably reproducing the experimental composition profiles. Insight into the composition- and temperature-dependence of various diffusion properties in fcc Ag–Sn–Zn alloy were subsequently retrieved and discussed. Furthermore, through the related Arrhenius information, the diffusion frequency factors and diffusion activity energies within the corresponding temperature and the composition range were obtained.

High-throughput determination of interdiffusivity in fcc Cu-Al-Sn and Cu-Ni-Al-Sn alloys

In this paper, 6 Cu-Al-Sn ternary and 8 Cu-Ni-Al-Sn quaternary solid diffusion couples within single fcc phase from 1023 K to 1173 K were fabricated. Employing the measured concentration profiles and interdiffusion fluxes, the atomic mobility parameters and interdiffusion coefficients in Cu-Al-Sn and Cu-Ni-Al-Sn systems were high-throughput evaluated using the numerical inverse method. The simulated composition distributions and interdiffusion fluxes in all the diffusion couples are consistent with the measured ones confirming the reliability of the atomic mobilities. The Matano-Kirkaldy method was also used to validate the reliability of the accuracy of the determined interdiffusion matrices. The 3D diagrams of main interdiffusivities in the Cu-Al-Sn and Cu-Ni-Al-Sn systems in the investigated ranges of composition and temperature were established to analyze the effects of components on the diffusivities. In the Cu-Al-Sn system, there is an increase in D˜AlAlCu with an increase in xAl, while D˜SnSnCu increases with increasing xSn but decreases with the addition of Al. Additionally, the introduction of Ni results in a decrease in all the main interdiffusivities (D˜NiNiCu, D˜AlAlCu and D˜SnSnCu) in the Cu-Ni-Al-Sn system.

Utilising physics-informed neural networks for optimisation of diffusion coefficients in pseudo-binary diffusion couples

ABSTRACT We propose a numerical inverse method based on physics-informed neural networks (PINN) for calculating composition-dependent diffusion coefficients in pseudo-binary (PB) diffusion couples in multicomponent alloys. Traditional methods rely solely on experimental diffusion profiles as design targets, which can lead to unreliable estimates. In contrast, PINN uses a combination of available data and physics-based constraints to obtain optimised design parameters and exact solutions for constrained governing differential equations. The constraints in PINN include governing partial differential equations, initial and boundary conditions, and any other equality/inequality relations obeyed by physical parameters. Our study shows the necessity of experimentally estimated intrinsic diffusion coefficients for the prediction of reliable composition-dependent mobility data. In the absence of such data, different combinations of unknown intrinsic diffusion coefficients can also produce reasonable approximations of diffusion profiles and interdiffusion coefficients while wrongly predicting more fundamental quantities (i.e. intrinsic diffusivities). Our method utilises PINN to simultaneously obtain optimised design parameters (diffusion coefficients) and exact solutions for governing diffusion equations. The implementation of PINN uses experimentally estimated diffusion coefficients obtained by the PB diffusion couple method, in addition to diffusion profiles, as design targets. The method is extended to non-ideal PB diffusion profiles, such as conventional diffusion couples in which all the components develop diffusion profiles, by incorporating additional constraints such as zero composition gradient of certain component(s) and experimentally estimated interdiffusion coefficients at extrema in the composition profiles. PINN is found to be a promising approach for obtaining reliable estimates of diffusion coefficients in multicomponent alloys.

Laser intensity and surface distribution identification at weld interface during laser transmission welding of thermoplastic polymers: A combined numerical inverse method and experimental temperature measurement approach

AbstractThe determination of the laser intensity profile necessary to weld two thermoplastic parts is crucial for understanding and optimizing the laser welding process in polymers and composite materials. Traditional methods of measuring the laser beam intensity profile, such as laser beam profiling techniques, can be difficult to implement and expensive. In this paper, we present the development of a technique that combines a numerical inverse method with experimental temperature measurements to determinate the laser intensity distribution at the weld interface. This technique does not require specialized or expensive equipment and accounts for variations in laser beam intensity caused by different components and interfaces. The technique uses thermocouple sensors to measure interface temperatures during welding, and an analytical gaussian model to estimate the laser intensity distribution at the weld interface. The model is then used as input data in a numerical heat transfer model. The technique is validated through experimental investigations on transparent and absorbent Polylactic acid polymers.

Diffusivity and atomic mobility for Fcc Ni–Ti–V alloys: Experiment and modeling

Ni–Ti–V is one key sub-system of the Ni-based superalloys. However, there is a lack of the investigation on diffusion kinetics for Fcc Ni–Ti–V alloys. In order to remedy this situation, we combined the diffusion couple experiment and calculation by CALTPP (CALculation of ThemoPhysical Properties) software to investigate diffusion kinetics for Fcc Ni–Ti–V alloys. Totally, 12 ternary Fcc Ni–Ti–V diffusion couples were prepared and annealed at 1273, 1373 and 1473 K, for which the composition profiles were measured by EPMA (Electron Probe MicroAnalysis). Based on the measured composition profiles and available thermodynamic descriptions, the interdiffusivities and atomic mobility parameters for Fcc Ni–Ti–V system were then calculated by the numerical inverse method integrated with CALTPP software. The simulated composition profiles by CALTPP software agree well with the measured ones. Moreover, the interdiffusivities obtained by CALTPP software were confirmed to be reliable through the good consistence with those calculated by M − K (Matano-Kirkaldy) method. Thus, the present numerical calculation by CALTPP software was verified to be reasonable. Finally, based on the presently obtained mobility parameters and available thermodynamic descriptions for Fcc Ni–Ti–V alloys, the three-dimensional surfaces of the interdiffusivities, activation energies and frequency-factors against Ti and V composition were predicted. The present work provides reliable kinetic parameters for Fcc Ni–Ti–V alloys, which can be included in the diffusion database of Ni-based superalloys for material design.

Accurate Modeling and Wave Propagation Analysis of Cracked Slender Structural Members by the Spectral Element Method

The analysis of elastic wave propagation in cracked structures is very useful in the crack detection by the ultrasonic guided wave method. This study presents an accurate spectral element modeling method for cracked slender structural members by using refined waveguide models and a more realistic crack model. Firstly, a spatial spectral beam element model is established for uncracked slender structural member based on the Love rod theory, the modified Timoshenko beam theory, and the Saint-Venant’s torsion theory. Then, the complete local additional flexibility matrix for crack in the structural member with rectangular cross section is derived from the theory of elastic fracture mechanics, and a two-node condensed spectral element model considering the stiffness coupling effect caused by the crack is established for cracked slender structural member. The wave response in cracked structures is solved by the numerical inverse Laplace transformation method. A thorough comparison of the wave responses in cracked structural member evaluated by the presented spectral element model and the 3D solid finite element model is given in the numerical example, which verifies the accuracy and high efficiency of the presented method.

High-throughput determination of interdiffusivity matrices and atomic mobilities in Cu-rich fcc Cu-Sn-Zn alloys by using the experimental composition profiles and HitDIC

In this paper, 12 groups of ternary Cu-Sn-Zn diffusion couples at 873, 973 and 1073 K were prepared and their composition profiles were measured. A high-throughput determination of atomic mobilities in Cu-rich fcc Cu-Sn-Zn alloys was performed by using HitDIC (HIgh-Throughput Determination of Interdiffusion Coefficients, https://hitdic.com/) software in the framework of the numerical inverse method. A unified set of atomic mobility parameters were thus acquired and excellent prediction performance towards all the composition profiles of the prepared diffusion couples was achieved. Good consistence between interdiffusion coefficients inferred by the assessed mobility descriptions and those evaluated with Matano-Kirkaldy method were successfully achieved. Reliability and soundness of evaluated mobility descriptions were therefore proved. Exploration of various diffusion properties including interdiffusivity and related Arrhenius information among the concerned temperature and composition range was also carried out. Insight into the composition and temperature-dependence of various diffusion properties in fcc Cu-Sn-Zn alloy were subsequently retrieved and discussed.

Investigation on the temperature-dependent diffusion growth of intermetallic compounds in the Mg-Al-Zn system: Experiment and modeling

With the rapid development of Mg alloys, deeper understanding to the thermodynamic and diffusional kinetic behavior of intermetallic compounds (IMCs) is important for studying the effect of alloying elements to the microstructure evolution. Specially, a systematic quantitative investigation on the diffusional growth of IMCs is of great necessity. However, the works studying the elemental diffusion behaviors of multiple-element IMCs are rare in magnesium alloy systems. The current work takes the ternary Mg-Al-Zn system as research target, and combines the diffusion couple technique, phase stability diagrams, in-situ observation technique and numerical inverse method to investigate the temperature-dependent kinetic coefficients. The parabolic growth constant (PGC) and interdiffusion coefficients for Mg solid-solution phase and γ-Mg17Al12, β-Mg2Al3, ε-Mg23Al30, MgZn2, Mg2Zn3, τ-Mg32(Zn, Al)49 and ϕ-Mg5Zn2Al2 IMCs in the Mg-Al-Zn alloy system are determined. By comparing the current experimental with calculation results, the rate-controlling factor of the temperature-dependent diffusion growth of ϕ, τ and ε ternary IMCs in the Mg-Al-Zn system is further discussed in detail.

High-throughput determination of the interdiffusion coefficients in fcc Cu–Ni–Al–Zn alloys

Nine solid state Cu–Ni–Al–Zn quaternary diffusion couples were prepared, from which the concentration distributions of Cu, Ni, Al and Zn at 1123, 1173, and 1223 K were determined by using the electron probe microanalyzer (EPMA) technique. Utilizing the measured concentration profiles, the mobilities in fcc Cu–Ni–Al–Zn system were obtained, from which the interdiffusion coefficients were determined in a high-throughput manner with numerical inverse method. It showed that the main interdiffusivities satisfy D˜ZnZnCu>D˜AlAlCu>D˜NiNiCu over the investigated composition and temperature ranges. The influences of individual components on the diffusion rates were evaluated based on the determined interdiffusivities. In addition, three-dimensional main interdiffusion coefficient maps were constructed. The effects of composition on the main diffusion coefficients were analyzed, showing that D˜AlAlCu and D˜ZnZnCu increase with the increasing Al and Zn additions but decrease with increasing Ni content, while D˜NiNiCu has a more complex trend as the concentrations of Al, Ni and Zn change. According to the obtained diffusion coefficients, ways to optimize the temperature and time during homogenization heat treatment of different series of Cu–Ni–Al–Zn alloys were determined.

Retroreflex ellipsometry for isotropic three-phase systems with nonplanar surfaces

Ellipsometry is a long-established and highly-accurate method for characterizing materials and thin films in the semiconductor, optical coating and biology industry. However, the geometric shape of samples is usually limited to flat samples for conventional ellipsometers because the operation must fulfill the condition of the law of reflection. The retroreflex ellipsometer developed at Fraunhofer IOSB can overcome the shape constraints without realignment of the sample but information on surface inclinations is still necessary. In this work, the polarization model of nonplanar surfaces is analyzed by polarization ray tracing and a numerical inverse method is proposed to calculate the angle of incidence for isotropic three-phase systems with known refractive indices and ellipsometric parameters (Ψ, Δ). The experiments show excellent accuracy for the measurements of the incident and tilt angles and prove that the method is capable of measuring the film thickness of nonplanar surfaces.

Formulation Method of Robot Kinematics Considering Versatility and Mobility of Each Joint

This paper proposes a method of formulating the robot kinematics. The proposed method can be applied to mobile manipulators, parallel link robots, and legged robots by the same procedure. The proposed forward kinematics formulation is based on tree-structured rigid-body system and is compared with the Denavit–Hartenberg method and URDF. The proposed differential kinematics formulation is represented by a block lower triangular matrix, which is a generalized form of the basic Jacobian matrix. In addition, an application example of the proposed formulation to the numerical inverse kinematics method is also shown. The usage of the proposed method is demonstrated by simulation experiments using a dual-arm mobile manipulator and a parallel link robot.

Stress analysis of adhesively-bonded single stepped-lap joints with functionally graded adherends based on the four-parameter fractional viscoelastic model

This paper presents a viscoelastic stress analysis of adhesively-bonded single stepped-lap joints under tension with functionally graded adherends. This analysis is based on the four-parameter fractional viscoelastic model and is considered the shear, peel, and axial stress components in the adhesive layer. The adhesive layer is assumed to exhibit a linear viscoelastic behavior, which is modeled using the fractional Zener model. The adherends is made of functionally graded Al2O3–Ni and are modeled using Timoshenko beam theory. The governing differential equations of the joint are derived in terms of the internal forces and moments in the Laplace domain using the structural, equilibrium, and compatibility equations of the adhesive layer and adherends. The governing equations are solved simultaneously and transformed from the Laplace domain into the time domain via the numerical inverse Laplace method. The results are compared to finite element results obtained from ANSYS Workbench, and a good agreement is observed. The results indicate that the axial stress in the adhesive layer is remarkable compared to the peel and shear stresses and must be considered in studies. Moreover, the variation in stress along the adhesive layer's thickness at both sides of the overlap region is significant.

High-throughput determination of mechanical and diffusion properties of Ti–Ta–Fe alloys

The mechanical and diffusion properties of Ti-Ta-Fe alloys in the Ti-rich region were investigated by utilizing a high-throughput method, with the combination of nanoindentation and diffusion couple techniques. Five groups of ternary Ti-Ta-Fe diffusion couples were prepared after annealing at 1273 K for 25 h. The composition-dependent mechanical properties of bcc Ti-Ta-Fe system were experimentally determined by means of nanoindentation and electron probe microanalysis (EPMA) techniques. Moreover, the interdiffusion coefficients of Ti-Ta-Fe alloys at 1273 K were confirmed from the composition gradients of the ternary diffusion couples with the support of a pragmatic numerical inverse method. A composition-dependent database on the mechanical and diffusion properties of Ti-Ta-Fe alloys was carefully established and utilized for the discussion of the processability during the hot working. The results indicated that the content of Fe should be controlled for the Ti alloys with high hardness and low Young's modulus.

Diffusion and Atomic Mobility for FCC Ni-Si-Ti Alloys: Experiment and Modeling

The Ni-Si-Ti system plays an important role in high temperature structural materials. Accurate atomic mobilities of fcc Ni-Si-Ti system are essential for the microstructure simulation of Ni-based superalloys. In the present work, diffusion couples of fcc Ni-Si-Ti system have been prepared and annealed at 1273, 1373 and 1473 K. By combining solid/solid diffusion couples with the electron probe microanalysis (EPMA) technique, the interdiffusivities in fcc Ni-Si-Ti were calculated via Matano-Kirkaldy method and numerical inverse method. Based on the reliable thermodynamic description and interdiffusion coefficients measured above, two sets of atomic mobilities were optimized by means of the CALTPP (CALculation of ThermoPhysical Properties) program. These atomic mobilities can reasonably describe the diffusion behaviors by comparing model-predicted diffusivities, concentration profiles and diffusion paths with the experimental ones. Besides, based on these atomic mobilities, three-dimensional surfaces for the interdiffusivity at 1273, 1373 and 1473 K were plotted. Furthermore, three-dimensional planes of the activation energy and frequency factor were evaluated using the Arrhenius equation.