A new inverse method for contaminant source identification under unknown solute transport boundary conditions

This work presents an application of the forward and inverse predictive modeling methodology of Cacuci and Ionescu-Bujor (2010) in the inverse mode to determine, within a tight a priori specified convergence criterion and overall accuracy, an unknown time-dependent boundary condition (specifically, the time-dependent inlet acid concentration) for a dissolver model case study by using measurements of the state function (specifically, the time-dependent acid concentration) at a specified location (specifically, in the dissolver’s compartment farthest from the inlet). The unknown time-dependent boundary condition is described by 635 unknown discrete scalar parameters. This forward and inverse predictive modeling methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution by using the a priori known mean values and uncertainties characterizing the model parameters, along with the computed and experimentally measured model responses and their covariances. This a priori distribution is subsequently combined using Bayes’ theorem with the likelihood provided by the computational model. The first-order response sensitivities serve as weighting functions in this objective combination of computational and experimental information.The use of the maximum entropy principle enables the forward and inverse predictive modeling of Cacuci and Ionescu-Bujor (2010) to construct an intrinsic regularizing metric for solving any inverse problem. In the present dissolver case study, the unknown time-dependent boundary condition is predicted by the methodology within an a priori selected convergence criterion, without user intervention and/or introduction of arbitrary regularization parameters, as the currently popular procedures need to do. This predictive modeling methodology yields optimally calibrated values for all model parameters, with reduced predicted uncertainties, as well as optimal (best-estimate) predicted values for the model responses (in this case study, the time-dependent acid concentrations in the dissolver’s compartments), also with reduced predicted uncertainties. Notably, even though the experimental data pertain solely to the compartment farthest from the inlet (where the data were measured), the application of this predictive modeling methodology actually improves the predicted values and reduces their predicted uncertainties not only in the compartment in which the data were actually measured but also throughout the entire dissolver, including the compartment farthest from the measurements (i.e., at the inlet). This is because this forward and inverse predictive modeling methodology combines and transmits information simultaneously over the entire phase-space, comprising all time steps and spatial locations.These results underscore the importance of this work in presenting the objective resolution (i.e., resolution in the absence of user-defined subjective adjustment of arbitrary regularization parameters) of a time-dependent inverse case study of potential importance to diversion activities associated with proliferation and international safeguards. The results obtained in this work establish confidence in the dissolver model’s accuracy for simulating the acid concentrations required to dissolve used nuclear fuel. In turn, these results will be used to generate source terms for key reprocessing facility components downstream and to support material accountability for nuclear safeguards.

Identification of material properties of hyper-elastic materials such as soft tissues of the human body or rubber-like materials has been the subject of many works in recent decades. Boundary conditions generally play an important role in solving an inverse problem for material identification, while their knowledge has been taken for granted. In reality, however, boundary conditions may not be available on parts of the problem domain such as for an engineering part, e.g., a polymer that could be modeled as a hyper-elastic material, mounted on a system or an in vivo soft tissue. In these cases, using hypothetical boundary conditions will yield misleading results. In this paper, an inverse algorithm for the characterization of hyper-elastic material properties is developed, which takes into consideration unknown conditions on a part of the boundary. A cost function based on measured and calculated displacements is defined and is minimized using the Gauss–Newton method. A sensitivity analysis is carried out by employing analytic differentiation and using the finite element method (FEM). The effectiveness of the proposed method is demonstrated through numerical and experimental examples. The novel method is tested with a neo–Hookean and a Mooney–Rivlin hyper-elastic material model. In the experimental example, the material parameters of a silicone based specimen with unknown boundary condition are evaluated. In all the examples, the obtained results are verified and it is observed that the results are satisfactory and reliable.

Simulating convective heat transfer using traditional numerical methods requires explicit definition of thermal boundary conditions on all boundaries of the domain, which is almost impossible to fulfill in real applications. Here, we address this ill-posed problem using machine learning techniques by assuming that we have some extra measurements of the temperature at a few locations in the domain, not necessarily located on the boundaries with the unknown thermal boundary condition. In particular, we employ physics-informed neural networks (PINNs) to represent the velocity and temperature fields while simultaneously enforce the Navier-Stokes and energy equations at random points in the domain. In PINNs, all differential operators are computed using automatic differentiation, hence avoiding discretization in either space or time. The loss function is composed of multiple terms, including the mismatch in the velocity and temperature data, the boundary and initial conditions, as well as the residuals of the Navier-Stokes and energy equations. Here, we develop a data-driven strategy based on PINNs to infer the temperature field in the prototypical problem of convective heat transfer in flow past a cylinder. We assume that we have just a couple of temperature measurements on the cylinder surface and a couple more temperature measurements in the wake region, but the thermal boundary condition on the cylinder surface is totally unknown. Upon training the PINN, we can discover the unknown boundary condition while simultaneously infer the temperature field everywhere in the domain with less than 5% error in the Nusselt number prediction. In order to assess the performance of PINN, we carried out a high fidelity simulation of the same heat transfer problem (with known thermal boundary conditions) by using the high-order spectral/hp-element method (SEM), and quantitatively evaluated the accuracy of PINN’s prediction with respect to SEM. We also propose a method to adaptively select the location of sensors in order to minimize the number of required temperature measurements while increasing the accuracy of the inference in heat transfer.

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition in the quasi-linear parabolic equation , with Dirichlet boundary conditions , , by making use of the over measured data and separately. The purpose of this study is to identify the unknown boundary condition at by using the over measured data and . First, by using over measured data as a boundary condition, we define the problem on , then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval , we reach the result. The main point here is the unique extensions of the solutions on to the closed interval which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition at .

Audio-frequency magnetotelluric (AMT) inversion technology plays a crucial role in the exploration of deep resources, especially in the detection of geothermal resources. Although AMT inversion technology has significant advantages in vertical resolution, the inversion process still faces serious challenges of nonuniqueness. To improve the accuracy and reliability of inversion, this paper develops an AMT data inversion method based on an adaptive binary structure constraint. This method incorporates adaptive extraction techniques to conduct a cluster analysis of the resistivity model generated in each iteration, thereby identifying and distinguishing the target and background areas in the model. These target areas are then used as prior structural information for the next iteration, guiding the refined reconstruction of the resistivity model. During the iteration process, the AMT inversion algorithm focuses on calculating the target areas while the resistivity values of the background areas remain unchanged. This strategy effectively reduces the degrees of freedom in the inversion problem and lowers the dimension of the solution, thereby improving the computational efficiency and accuracy of the inversion. Model tests show that compared with the traditional smoothing-constrained AMT (S-AMT) inversion method, the adaptive binary structure-constrained AMT inversion method (ABS-AMT) can reconstruct high-resolution refined resistivity models, significantly reducing the uncertainty of inversion. Furthermore, the ABS-AMT method not only reduces the dependence on prior knowledge but also enhances the algorithm’s adaptability to different geologic conditions. Practical application in the geothermal area of Laosandui in Linjiang City, Jilin Province, China, fully demonstrates the method’s effectiveness in known geothermal areas.

This paper focuses on transverse moisture transport in spruce wood caused by changing climate conditions. Aiming to determine the space and time dependent moisture content (MC) field in the cross section of the sample during the drying process, at first a series of measurements on small clear specimens of spruce wood were carried out and secondly a computer simulation of the drying experiment was done. The MC was measured by the destructive gravimetric method. The measurement was repeated several times for consecutive time intervals during the drying period. In a computer aided material characterization, a so-called inverse problem identification method is used. The solution of the inverse problem enabled us to estimate the moisture diffusion coefficients and to simultaneously determine the MC field.

Inverse estimation of the unknown heat flux boundary with irregular shape fins

This paper presents an inverse analysis method for the identification of unknown boundary condition in elastoplastic state, using a finite element method (FEM) on a small memory capacity machine such as a microcomputer and a personal computer. This inverse analysis method is composed of following two calculation processes: an identification process which obtains a solution in elastic state, and a direct analysis in elastoplastic state based on the identified results from. For efficient identification, uses a penalty function method and an attack technique based on both accelerative moving lower and upper bounds. A zooming identification method is available for improvement of the identification accuracy. adopts a triple-tandem-type substructuring method as an efficient calculation method. Therefore, we establish an efficient inverse analysis method for the identification of unknown boundary condition in elastoplastic state. Finally, the validity of the inverse analysis method is confirmed from the numerical solution of a three-dimensional finite element model, as an example of V-shaped grooved structure with reinforced members under tension, executed on a small machine.

Porous media represent a vast class of materials, encountered in, i.e., geology, medicine and acoustics. Knowledge of the porous microstructure is necessary for the use of the different models describing the acoustical behavior of porous media. The focus of this work is put on inverse identification methods, that allow the retrieval of several microstructural parameters of a porous material. The inverse method is based on the measurement of an acoustic field transmitted and/or reflected by the porous material, and is recast as a Bayesian inference problem. This allows for an uncertainty quantification of the identified parameters, and model selection, in case of competing models for the acoustic representation of a porous material. Different materials are tested in impedance tubes, and their properties are inferred using different acoustical models. Samples of multiple thicknesses are backed with airgaps of different length, which allows for a densification of the test matrix for a given material: this ultimately improves knowledge on the porous microstructure properties, described by posterior probability density functions in a Bayesian context.

An inverse method based on the symbolic approach is proposed to determine the boundary condition in three-dimensional inverse heat conduction problems. This method uses symbols to represent the unknown boundary condition and then executes an implicit alternating-direction finite-difference method to calculate the temperature distribution. The calculated results are expressed explicitly as functions of the symbol of the undetermined boundary condition. Then, a set of linear equations is constructed from the comparison between the output symbolic temperature and the measured numerical temperature. Thus, the set of linear equations is solved through the linear least-squares error method and leads to the solution of the unknown boundary condition. Results from examples confirm that the proposed method is applicable in solving the three-dimensional inverse heat conduction problem. In the example problems, a temporally dependent and a spatially - temporally dependent boundary condition are used to demonstrate the procedure of the proposed method. In the first example, the temporally dependent boundary conduction, the result shows that only one-point measurement is needed to estimate the temporally dependent condition whether the measurement errors are considered or not. In the second problem, the four- and eight-point measurement methods are adopted to estimate the temporally - spatially dependent boundary condition. The result shows that only the four-point measurement is needed to estimate the surface temperature when measurement errors are neglected. When measurement errors are considered, the eight-point measurement is required in order to increase the congruence of the estimated results to the exact solutions.

The present paper aims to assess the accuracy of identification methods used in the evaluation of the flow stress relationship of tubular materials for hydroforming applications. Based on experimental data acquired from home designed and manufactured experimental tool and results collected from literature, flow stress parameters are determined using both analytical and inverse identification methods. The obtained results are coped to experimental measurements to validate the proposed approaches. It is shown from the analysis based on the comparative assessment of flow stress inferred from tube bulge test that, inverse parameter identification method is the appropriate methodology that contribute to a more accurate tube hydroforming characterization.

Two accurate finite-difference frequency-domain (FDFD)-based iterative inverse methods, referred to as FDFD-based iterative method (FIM) and distorted FDFD-based iterative method (DFIM), are proposed for electromagnetic inverse scattering. They can be considered as FDFD-based versions of BIM and DBIM and inherit the advantage of the traditional direct FDFD inversion method, i.e., there is no need for Green’s function of complex background, especially for the inhomogeneous medium. The difference is that full-wave consideration is implemented in the proposed methods to build an iterative framework. Then, the scattered field and object parameter profile are updated in each iteration. Using the high-order components in iteration formulation, both the methods are able to reconstruct high-contrast objects and provide accurate reconstruction results. Compared with other full-wave methods based on differential equation, the two methods also have a distinct characteristic that the inversion accuracy is not limited by extra constraints. The difference between the two proposed methods is whether the background coefficient matrix is updated. Generally, DFIM has faster convergence than FIM due to the updated background coefficient matrix in the iterations. To demonstrate the effectiveness of both the methods, several typical 2-D experiments are conducted. The results show that the proposed methods could achieve good accuracy and high imaging quality.

The coupled heat conduction, convection and radiation problem for three heated insulated cables suspended in air is solved by an operator-splitting pseudo-time-stepping finite element method, which automatically satisfies the continuity of the interfacial temperature and heat flux. The main feature of the solution procedure is that the multi-phases are treated as a single computational domain with unknown interfacial boundary conditions. The temperature distribution in the heated metal cylinder, in the insulating layer and in the open air together with the convective flow pattern are obtained simultaneously.

An inverse boundary value problem for the Helmholtz equation in a bounded domain is considered. The problem is to extract information about an unknown obstacle embedded in the domain with unknown impedance boundary condition (the Robin condition) from the associated Dirichlet-to-Neumann map. The main result is a characterization of the unknown obstacle via the sequences that are constructed by the Dirichlet-to-Neumann map, under smallness conditions on the wave number and the upper bound of the impedance. Moreover two alternative simple proofs of a previous result of Cheng-Liu-Nakamura which are based on only some energy estimates, an analysis of the blowup of the energy of so-called reflected solutions and an application of the enclosure method to the problem are also given.

The paper deals with the determination of the symmetric model and proper boundary conditions for solving nonlocal elastic symmetric structures. The above concepts, in the context of nonlocal integral elasticity, turn out to be different with respect to the standard ones, classically applied when dealing with local elastic symmetric structures. Indeed, when only a symmetric portion of the structure is analyzed, the nonlocal effects induced by the remaining (cut) portions are lost, this necessitates the consideration of an enlarged symmetric model on which appropriate nonlocal boundary conditions have to be imposed. It has to be pointed out how the width of such an enlarged model depends on the nonlocal material parameters, while the correct unknown nonlocal boundary conditions are here obtained and enforced by an iterative procedure. The accuracy of the proposed approach in solving nonlocal structural symmetric problems is tested with the aid of two numerical examples.