Numerical Inverse Method Research Articles

On the temperature-dependent diffusion growth of ϕ-Mg5Al2Zn2 ternary intermetallic compound in the Mg–Al–Zn system

The study on the temperature-dependent kinetic behavior of intermetallic compounds (IMCs) in Mg–Al–Zn (AZ) series alloy system is important since its close interrelation with the microstructure evolution under various alloying conditions as well as the T-dependent performance. However, there is a very lack of experimental study on the diffusion kinetics of the ternary IMCs in the AZ series alloy systems. The current work combines the diffusion couple technique and numerical inverse method to investigate the T-dependent kinetic coefficients, i.e., the parabolic growth constant (PGC) and interdiffusion coefficients, for ϕ-Mg5Al2Zn2 ternary IMC in the Mg–Al–Zn alloy system. The Arrhenius formula of both PGC and main interdiffusivities is obtained, i.e., $${}_{{}}^{\phi } k_{p} = 5.48 \times 10^{ - 5} \exp \left( {\frac{120010}{{RT}}} \right)$$ m2/s, $${}_{{}}^{\phi } \tilde{D}_{{{\text{MgMg}}}}^{Al} = 3.48 \times 10^{ - 10} \exp \left( {\frac{50420}{{RT}}} \right)$$ m2/s and $${}_{{}}^{\phi } \tilde{D}_{{{\text{ZnZn}}}}^{Al} = 2.87 \times 10^{ - 7} \exp \left( {\frac{91440}{{RT}}} \right)$$ m2/s, based on which a numerical reproduction of the experimentally determined IMC growth is performed for verification. By comparing the current experimental and calculation results, the rate-controlling factor of the temperature-dependent diffusion growth of ϕ ternary IMC in the Mg–Al–Zn system is further discussed.

Transient analysis of piecewise homogeneous QBD process

The article presents a numerical analysis approach for the transient solution of a piecewise homogeneous quasi-birth-death process. The proposed approach computes the transient probabilities based on the linear combination of matrix geometric series in Laplace transform domain, and builds on the availability of an efficient numerical inverse Laplace transformation method.

Thermospectroscopic infrared imaging of a confined drying process

We present an infrared (IR) imaging technique that allows us to retrieve quantitative concentration and thermal maps with relatively fast acquisition times for samples that are evolving in time and have micron-scale spatial resolution. As a proof-of-concept, we image the transient drying kinetics of a μL drop of colloidal suspension in a confined geometry. Quantitative concentration maps inside the drying droplet are retrieved. Transport phenomena such as colloid redistribution inside the droplet due to inhomogeneous drying can be highlighted by this means. A numerical inverse method based on the acquired images that allows one to estimate intrinsic properties of the studied material, such as the collective diffusion coefficient of the mixture, is presented. Such a technique combined with statistical inverse methods provides a useful, non-invasive means of visualizing and estimating parameters of materials evolving in time.

Atomic mobilities and diffusivities in fcc_A1 Ni–Cr–V system: Modeling and application

Ni–Cr–V system is a significant sub-system of both cemented carbides and superalloys. In the present work, the atomic mobilities for fcc_A1 Ni–Cr–V alloy were evaluated by means of the CALTPP (CALculation of ThermoPhysical Properties) program, which utilizes the thermodynamic description and the interdiffusivity obtained in our previous work by a numerical inverse method. The reasonability was validated through comparing the CALTPP-predicted composition profiles, interdiffusivities and diffusion paths with the experimentally measured ones. Furthermore, in order to study the diffusion behaviors, the present atomic mobilities were applied to calculate the diffusion fluxes, the interdiffusivities over the whole investigated composition and temperature ranges, the pre-frequency factors as well as the activation energy. It can be seen from the 3-dimensional interdiffusivity planes that when the composition of Cr is more than that of V, the diffusion of Cr is generally faster than V. The pre-frequency factor indicates that the frequency of collisions between Cr atoms is higher than that of V while the activation energy shows that the diffusion of Cr requires more energy than V. The presently obtained diffusion characteristics of fcc_A1 Ni–Cr–V alloys are critical for predicting and controlling the microstructure evolution and further designing high-performance cemented carbides as well as superalloys.

Diffusion growth of ϕ ternary intermetallic compound in the Mg-Al-Zn alloy system: In-situ observation and modeling

Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property. However, there is a very lack of existing data due to difficulties in both experimental and computational aspects. The current work aims at presenting the experimental observation on the diffusion growth behavior of ϕ phase at 360 °C as well as calculating its composition-dependent interdiffusion coefficients. We designed and successfully fabricated four Mg-τ ternary diffusion couples annealed at 360 °C for different times, where the diffusion path goes across the ϕ phase region and the diffusion growth of ternary intermetallic compound can be solely detected. In-situ observation of the time-dependent growth of ϕ phase was performed to accurately determine the parabolic growth constant. The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds, which can reproduce the presently observed diffusion growth behavior of ϕ ternary intermetallic compound in Mg-τ diffusion couples.

Diffusivity and atomic mobility for fcc Ni–Cu–Ti alloy: Measurements and an intelligent modeling

The diffusivities of fcc Ni-rich Ni–Cu–Ti alloy have been systematically investigated by means of experimental measurements and an intelligent modeling. Utilizing two bulk diffusion couples without intersection points in diffusion paths together with electron probe microanalysis technique, the interdiffusion coefficients along the whole diffusion path in fcc Ni–Cu–Ti alloy at 1273, 1373 and 1473 K are determined by a novel numerical inverse method. Three new diffusion couples together with two initial diffusion couples are used to obtain the diffusivities by Matano-Kirkaldy method. Two sets of atomic mobilities are established based on the interdiffusion coefficients obtained using numerical inverse method and Matano-Kirkaldy method, respectively, through an intelligent modeling in CALTPP (CALculation of ThermoPhysical Properties) program. The reliability of present two sets of atomic mobilities is verified by comprehensive comparisons between the calculated and experimental properties, including diffusivities, concentration profiles and diffusion paths. A preferable agreement is found in the comparisons where the diffusivities are calculated using numerical inverse method. The present investigation of diffusivities for fcc Ni–Cu–Ti alloy demonstrates that two diffusion couples combined with the numerical inverse method can establish a reliable atomic mobility for the targeted alloys.

High-throughput determination of high-quality interdiffusion coefficients in metallic solids: a review.

Accurate interdiffusion coefficients of composition and temperature dependence are significantly important for understanding different materials processes. However, the high-throughput determination of high-quality interdiffusion coefficients, especially in multicomponent systems, has been sustaining as a challenge in materials community. This review dealt with a comprehensive summarization of the recent progress in this field, aiming at advancing a scientific routine for realizing the high-throughput determination of high-quality interdiffusion coefficients in metallic solids. First, an introduction of traditional Matano-based approaches and their recent development was given. Second, the numerical inverse methods were described, with a focus on the recently developed pragmatic numerical inverse method and related public toolkits. Potential strategies for resolving the problems about accuracy and uniqueness of the solutions to the numerical inverse methods were highlighted. The combination of numerical inverse method and diffusion multiple technique was highly proposed for high-throughput determination of interdiffusion coefficients in metallic solids with any number of components. After that, the case studies on the high-throughput determination of interdiffusivity matrices in the real Ni-based, high-entropy/high-entropy superalloys were demonstrated. Discussion on the substitution of Re in Ni-based single-crystal superalloys and the sluggish diffusion in high-entropy/high-entropy superalloys was also carried out. Fourth, the general idea for the uncertainty quantification was proposed in order to obtain high-quality interdiffusion coefficients, followed by the introduction of recent progress on the uncertainty quantification in both Matano-based methods and numerical inverse methods. Finally, the conclusions were drawn, and the future trends in diffusion community were also pointed out.

Analytical Model for Leakage Detection in CO2 Sequestration in Deep Saline Aquifers: Application to ex Situ and in Situ CO2 Sequestration Processes.

One of the prominent methods for carbon dioxide sequestration is disposal into deep saline aquifers. This is mainly because deep saline aquifers provide significant capacity for storage of unwanted fluids underground for a long period. However, saline aquifers may have a leaky cap rock. The sealing capacity of a cap rock must, therefore, be evaluated to ensure the integrity and safety of its storage media; hence robust classifications of the cap rock are required even before starting the storage/disposal operations. Aqueous fluids can be injected into a target storage aquifer, and pressure changes owing to leakage can be monitored in an upper aquifer separated by a cap rock for a short period. The measurement of pressure responses in the monitoring aquifer can be used to identify and characterize any leakage path in the cap rock. This paper provides analytical models in the Laplace domain for both in situ and ex situ CO2 sequestration methods. Using the numerical Laplace inverse method called the “Stehfest” method, the analytical solution in the real domain is calculated. The analytical solutions developed can be used for determining both dimensionless pressure changes in monitoring and storage aquifers due to leakages and dimensionless leakage rates.

Identification of the material behavior of adhesive joints under dynamic multiaxial loadings

This paper presents a numerical inverse method dedicated to the characterization of adhesive joints under multiaxial and dynamic loading conditions. The properties under scrutiny are the constitutive behavior of the joint as well as the final fracture surface. The experimental setup consists of a Split Hopkinson Pressure Bar system and local strain measurements performed by Digital Image Correlation (DIC) as well as a novel specific sandwich specimen denoted as DODECA. The direct numerical model is an original Finite Element computation combining 3D and 1D elements for an optimal handling of wave reflection and interfaces. It further provides an optimal compromise between computation time and accuracy. The identification method is based on the Finite Element Model Updating method (FEMU). Material parameters are identified for three different multiaxial loading conditions and presented as yield and fracture surfaces in the space of equivalent von Mises stress vs. hydrostatic stress. As a complement of the analysis, uncertainties and confidence of the identified parameters are estimated with precise qualitative tools.

Determination of accurate interdiffusion coefficients in fcc Ag-In and Ag-Cu-In alloys: A comparative study on the Matano method with distribution function and the numerical inverse method with HitDIC

In this paper, 3 fcc binary Ag-In and 3 fcc ternary Ag-Cu-In diffusion couples were firstly prepared. Based on the measured composition-distance profiles, both the traditional Matano method with distribution functions and the numerical inverse method with HitDIC were then utilized to determine the accurate interdiffusion coefficients in fcc Ag-In alloys at 873 K, 973 K, and 1073 K and Ag-Cu-In alloys at 1073 K. The determined interdiffusivities in fcc Ag-In alloys increases as the composition of In increases. The impurity diffusivities of In in fcc Ag extrapolated from the evaluated interdiffusivities based on the Matano method agree well with the tracer experimental data from the literature. As for fcc Ag-Cu-In alloys, the three-dimensional main interdiffusivities planes at 1073 K were constructed over the investigated composition range. The diffusion rate of In is generally faster than that of Cu. Moreover, the main interdiffusivity of In increases with the increment of both In and Cu concentrations, while the main interdiffusivity of Cu increases with In concentration but decreases with Cu concentration. After that, a comprehensively comparative study on the two methods were performed. The interdiffusivities in both the fcc Ag-In and Ag-Cu-In alloys evaluated from the two methods are consistent with each other, indicating that both methods can result in accurate interdiffusivities in binary and ternary alloys. However, as the number of elements increases, the Matano method becomes inconvenient. While for the numerical inverse method with HitDIC, no such limitation exists, and it can be conveniently applied in alloys with any number of components.

On diffusion behaviors in face centered cubic phase of Al-Co-Cr-Fe-Ni-Ti high-entropy superalloys

Owning the excellent high temperature properties, low cost and density, Al-Co-Cr-Fe-Ni-Ti high-entropy superalloys (HESAs) represent a potential substitute for Ni-based superalloys. In this paper, the composition-dependent interdiffusivity matrices of face centered cubic (fcc) Al-Co-Cr-Fe-Ni-Ti HESAs at 1473 K were first determined via a combination of diffusion multiple technique and HitDIC software based on the numerical inverse method. The correlations between the main interdiffusivities and the components suggest that alloying elements play great impact on diffusion in HESAs. After that, comprehensive comparisons of the interdiffusivities and tracer diffusivities in present fcc HESAs with the literature data in fcc-based pure elements, lower-order alloys and traditional high-entropy alloys were conducted. The results indicate that the sluggish diffusion effect exists for only some of the components in the present HESAs. Moreover, comparisons of the diffusion coefficients between the HESAs and traditional Ni-based superalloys with refractory elements show that the interdiffusivities in the present HESAs are lower than the Ni-based superalloys with most refractory elements, but higher than those with Ir, Re and Os. Furthermore, it is also found that refractory elements need to be added in order to further improve the creep resistance of the HESAs.

Experimental Investigations on the Quaternary Interdiffusion Coefficients, Young’s Modulus and Hardness in bcc Ti-Nb-Ta-Zr Quaternary Alloys

Accurate interdiffusion coefficient, Young’s modulus and hardness are essential for precisely controlling the production of a homogeneous bio-metallic alloys with excellent properties. In this work, we utilized a combinational approach by combining the advanced diffusion multiple technique and the pragmatic numerical inverse method to efficiently determine the composition–dependent interdiffusion coefficient matrices in bcc quaternary Ti-Nb-Ta-Zr alloys at 1373 K. Moreover, the composition-dependent Young’s modulus and hardness of bcc Ti-Nb-Ta-Zr alloys were obtained by means of the nanoindentation technique.

Errors of Misclassification Associated with Edgeworth Series Distribution (ESD)

This study investigates the errors of misclassification associated with Edgeworth Series Distribution (ESD) with a view to assessing the effects of sampling from non-normality. The effects of applying a normal classificatory rule when it is actually a persistent non-normal distribution were examined. These were achieved by comparing the errors of misclassification for ESD with ND using small sample sizes at every level of skewness factor. The simulation procedure for the experiment of the study was implemented using numerical inverse interpolation method in R program to generate a uniformly distributed random variable N. A configuration size of 1000 was obtained for the two training samples drawn at every level of skewness factor (λ3), in the range (0.00625, 0.4). This was repeated for different small sample sizes by comparing errors of misclassification of ESD with ND. The simulation results showed that the optimum probabilities of misclassification by ESD: (E12E) decreases and (E12E) increases, as the skewness factor (λ3) increases. The optimum total probability of misclassification is stable as λ3 also increases. The probability of misclassification E12E ≥E12N and E21E ≥E21N at every level of λ3. Thus, the total probabilities of misclassification are not greatly affected by the skewness factor. This asserts that the normal classification procedure is robust against departure from normality.

Numerical inverse method for calculating heat flux in temperature-sensitive-coating measurement on a finite base

The one-dimensional unsteady heat transfer of a polymer layer on a finite base is studied. A numerical inverse method is developed to calculate heat flux from surface-temperature images obtained using temperature-sensitive coating particularly in high-enthalpy wind tunnels. It is found that the effect of the finite thickness of the base is significant as time increases such that the semi-infinite base assumption made in many methods is no longer accurate in heat-flux calculation. The method developed in this work can take into account the effects of the finite thickness of a base and the temperature-dependencies of the thermal properties of the materials. This method has been validated through simulations and processing the experimental temperature-sensitive-paint (TSP) images.

Mechanical Performance of Resistance Spot Welded Steel Sheets

Mechanical performance of resistance spot welded steel sheets was evaluated under quasi-static and dynamic loading conditions. Numerical inverse calibration method was mainly utilized to characterize mechanical properties especially for flow curves and ductile fracture criteria of constituent zones of the welded joints. For the base sheets, standard tensile test and high-speed tensile test was conducted under quasi-static and dynamic loading conditions, respectively. Mechanical properties of the weld zones were characterized using the developed miniature tensile specimen. Implementing the mechanical properties and structure to the finite element model, finite element simulation was used to evaluate the mechanical performance of two distinct weld coupons.

Out-plane stability safety factors of CFST arches using inverse finite element reliability method

Concrete filled steel tubular arch (CFST) bridges have been widely used in China since the 1990s. However, researches on the stability design of this type of bridges are limited, especially for the out-plane stability. Therefore, a numerical inverse finite element reliability method is presented to solve the problem. In this method, safety factors are introduced into the limit state equations. The cooperation of programming and general finite element software is utilized to implement the method. For different target reliability indices, out-plane stability safety factors of CFST arches considering nonlinearities are obtained via the method. The results show that: the method is of good efficiency and applicability; steel ratios and lateral loads have significant influences on the safety factors; the effects of resistance and load uncertainties on the factors are important which should be concerned in stability design of CFST arch bridges.

3D-specific absorption rate estimation from high-intensity focused ultrasound sonications using the Green's function heat kernel.

To evaluate a numerical inverse Green's function method for deriving specific absorption rates (SARs) from high-intensity focused ultrasound (HIFU) sonications using tissue parameters (thermal conductivity, specific heat capacity, and mass density) and three-dimensional (3D) magnetic resonance imaging (MRI) temperature measurements. SAR estimates were evaluated using simulations and MR temperature measurements from HIFU sonications. For simulations, a "true" SAR was calculated using the hybrid angular spectrum method for ultrasound simulations. This "true" SAR was plugged into a Pennes bioheat transfer equation(PBTE) solver to provide simulated temperature maps, which were then used to calculate the SAR estimate using the presented method. Zero mean Gaussian noise, corresponding to temperature precisions between 0.1 and 2.0°C, was added to the temperature maps to simulate a variety of invivo situations. Experimental MR temperature maps from HIFU sonications in a gelatin phantom monitored with a 3D segmented echo planar imaging MRI pulse sequence were also used. To determine the accuracy of the simulated and phantom data, we reconstructed temperature maps by plugging in the estimated SAR to the PBTE solver. In both simulations and phantom experiments, the presented method was compared to two previously published methods of determining SAR, a linear and an analytical method. The presented numerical method utilized the full 3D data simultaneously, while the two previously published methods work on a slice-by-slice basis. In the absence of noise, SAR distribution estimates obtained from the simulated heating profiles match closely (within 10%) to the initial true SAR distribution. The resulting temperature distributions also match closely to the corresponding initial temperature distributions (<0.2°C RMSE). In the presence of temperature measurement noise, the SAR distributions have noise amplified by the inverse convolution process, while the resulting temperature distributions still match closely to the initial "true" temperature distributions. In general, temperature RMSE was observed to be approximately 20-30% higher than the level of the added noise. By contrast, the previously published linear method is less sensitive to noise, but significantly underpredicts the SAR. The analytic method is also less sensitive to noise and matches SAR in the central plane, but greatly underpredicts in the longitudinal direction. Similar observations are made from the phantom studies. The described numerical inverse Green's function method is very fast - at least two orders of magnitude faster than the compared methods. The presented numerical inverse Green's function method is computationally fast and generates temperature maps with high accuracy. This is true despite generally overestimating the true SAR and amplifying the input noise.

Influence of dynamic loading on failure behavior of spot welded automotive steel sheets

Dynamic failure of the spot welded automotive steel sheets, similar and dissimilar combinations of TRIP980 and GMW2 sheets, was analyzed with finite element (FE) simulations. The strain rate-sensitivity of the base sheets was considered to investigate the influence of dynamic loading on failure behavior of the welded sheets. Mechanical properties were first identified under the quasi-static loading as the reference condition. Hardening behavior and fracture criterion were identified by numerical inverse method applied to the standard tensile test and a miniature tensile test for the base sheet and the weld nugget, respectively. Moreover, strain rate-sensitivities of the base sheets were inversely characterized as a function of strain and strain rate. Dynamic lap-shear and coach-peel coupon tests were performed to evaluate the dynamic failure behavior of the welded sheets. Peak strength, failure mode and load-displacement curves of the coupon tests were predicted with the FE simulations. The combined experimental and numerical approach showed that the strain rate-sensitivity of the base sheet played a key role on determination of failure characteristics in coupon tests.

Simulation of Student–Lévy processes using series representations

Levy processes have become very popular in many applications in finance, physics and beyond. The Student–Levy process is one interesting special case where increments are heavy-tailed and, for 1-increments, Student t distributed. Although theoretically available, there is a lack of path simulation techniques in the literature due to its complicated form. In this paper we address this issue using series representations with the inverse Levy measure method and the rejection method and prove upper bounds for the mean squared approximation error. In the numerical section we discuss a numerical inversion scheme to find the inverse Levy measure efficiently. We extend the existing numerical inverse Levy measure method to incorporate explosive Levy tail measures. Monte Carlo studies verify the error bounds and the effectiveness of the simulation routine. As a side result we obtain series representations of the so called inverse gamma subordinator which are used to generate paths in this model.

Screening of Possible Re-Substitutional Elements in Single-Crystal Ni-Based Superalloys: A Viewpoint From Interdiffusion Coefficients in Ni-Al-X Ternaries

A popular area of research in the field of high-temperature alloys concerns the search of substitutional elements for Re in order to manufacture single-crystal Ni-based superalloys with less or even no Re addition. To find the elements with similar or even lower diffusion coefficients than Re is an effective strategy. Based on 29 fcc diffusion couples in ternary Ni-Al-X (X = Re, Os, and Ir) systems, high-throughput measurement of composition- and temperature-dependent interdiffusivity matrices was performed using our recently developed numerical inverse method implemented in HitDIC software. The reliability of the determined interdiffusivities was validated by comprehensively comparing the model-predicted composition/interdiffusion flux profiles for each diffusion couple with the corresponding experimental data. Moreover, we also conducted a comparison with the interdiffusivities evaluated using the traditional Matano–Kirkaldy method as well as those from the literature and in boundary binary systems. After that, a comprehensive comparison of the interdiffusion coefficients in fcc Ni-2 wt pct Al-6 wt pct X (X = Ti, Co, Ni, Nb, Mo, Ru, Rh, Ta, W, Re, Os, Ir, and Pt) alloys at 1423 K to 1573 K was conducted. Results indicate that the diffusion rate of Re is lower than that of Os at 1473 K and 1523 K; but higher at 1573 K, while the diffusion rate of Ir is always slightly higher than those of Os and Re at 1473 K to 1573 K. Further analysis of the magnitude of the interdiffusion coefficient correlates with the alloying concentration, activation energy, atomic number, and atomic radius of different diffusing transition metal species (i.e., Ti, Co, Ni, Nb, Mo, Ru, Rh, Ta, W, Re, Os, Ir, and Pt) was conducted, which is expected to provide useful information regarding element choice in the development of new-generation Ni-based single-crystal superalloys.