Source Boundary Conditions Research Articles

Simultaneous identification of groundwater contamination source information, model parameters, and boundary conditions under an unknown boundary mode

Simultaneous identification of groundwater contamination source information, model parameters, and boundary conditions under an unknown boundary mode

Finite element modeling and simulation of IN738 laser powder bed fusion process assisted via machine learning

Finite element modeling and simulation are considered one of the effective methods for optimizing additive manufacturing process windows, which can better capture metallurgical features under the interaction of multiple physical fields. However, correctly selecting or obtaining heat source parameters, boundary conditions, and thermophysical parameters is the biggest challenge of finite element modeling. In this work, the laser semi-ellipsoid heat source parameters and powder thermophysical parameters of the laser powder bed fusion process was effectively determined by machine learning method, and an accurate laser powder bed fusion process model of IN738 alloy was established. The simulation error of the size of single-track molten pool under different process conditions is less than 2.0 %. The simulation results of stress field by multi-layer and multi-track under different process conditions are consistent with the micro crack and density test results of deposited samples. The relevant research results provide a new idea for correctly establishing the model of additive manufacturing process and realizing reliable finite element simulation.

General decay results for a viscoelastic wave equation with the logarithmic nonlinear source and dynamic Wentzell boundary condition

In this work we investigate a viscoelastic wave equation involving a logarithmic nonlinear source and dynamic Wentzell boundary condition. Making some assumptions on the memory kernel function and using convex function theory and Lyapunov method, we establish the general decay estimate of the solutions. Finally we give two examples to illustrate our results.

Discovering physics-informed neural networks model for solving partial differential equations through evolutionary computation

In recent years, the researches about solving partial differential equations (PDEs) based on artificial neural network have attracted considerable attention. In these researches, the neural network models are usually designed depend on human experience or trial and error. Despite the emergence of several model searching methods, these methods primarily concentrate on optimizing the hyperparameters of fully connected neural network model based on the framework of physics-informed neural networks (PINNs), and the corresponding search spaces are relatively restricted, thereby limiting the exploration of superior models. This article proposes an evolutionary computation method aimed at discovering the PINNs model with higher approximation accuracy and faster convergence rate. In addition to searching the numbers of layers and neurons per hidden layer, this method concurrently explores the optimal shortcut connections between the layers and the novel parametric activation functions expressed by the binary trees. In evolution, the strategy about dynamic population size and training epochs (DPSTE) is adopted, which significantly increases the number of models to be explored and facilitates the discovery of models with fast convergence rate. In experiments, the performance of different models that are searched through Bayesian optimization, random search and evolution is compared in solving Klein–Gordon, Burgers, and Lamé equations. The experimental results affirm that the models discovered by the proposed evolutionary computation method generally exhibit superior approximation accuracy and convergence rate, and these models also show commendable generalization performance with respect to the source term, initial and boundary conditions, equation coefficient and computational domain. The corresponding code is available at https://github.com/MathBon/Discover-PINNs-Model.

Joint identification of groundwater pollution source information, model parameters, and boundary conditions based on a novel ES-MDA with a wheel battle strategy

Groundwater pollution source identification (GPSI) is an important prerequisite for pollution remediation and risk assessment. However, an accurate GPSI is usually difficult to achieve due to the need to simultaneously consider practical problems and theoretical research. For practical problems, the boundary conditions, especially the concentration boundary, are often given as known constants through prior information. However, in most practical situations, the boundary conditions are complex and cannot be accurately estimated in advance, which leads to the distortion of the final identification results. Therefore, this study focused on the concentration boundary, and first proposed to jointly identify three types of unknown variables (source information, model parameters, and boundary conditions) to ensure that the identification results had more practical value. For theoretical research, as the number of unknown variable types increases, the difficulty of solving the inverse problem often increases, which may lead to inaccurate inversion results. Thus, a novel ensemble smoother with multiple data assimilation (ES-MDA) with a wheel battle strategy was proposed, enhancing the identification accuracy. We designed two cases to verify the effectiveness and practicality of the above ideas: a low-dimensional Case 1 (including four different synthetic scenarios: three scenarios with unknown boundary conditions under three concentration boundary modes and a scenario with known boundary conditions) and a high-dimensional complex Case 2. Identifying the boundary conditions in GPSI was found to be of great significance. Compared to the standard ES-MDA method, the ES-MDA with a wheel battle strategy proposed here could improve the inversion accuracy, and had certain effectiveness and practicality.

On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions

We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.

An offline coupling approach for efficient SPH simulations of long-distance tsunami events using wave source boundary condition

The entire procedure of tsunami events consists of wave generation and propagation toward the coastal structures encompassing the long-range system domain. Therefore, instead of an integrated continuous simulation, two or several consecutive simulations are preferred for numerical efficiency, where the hydrodynamic information within a particular domain from the preceding simulation can be utilized for the wave sources in the subsequent simulation. In this study, as an offline coupling approach, the wave source boundary condition is proposed based on the particle-based Lagrangian framework. This boundary condition is integrated with the solution algorithm of the smoothed particle hydrodynamics (SPH) in-house parallel code. In the dambreak simulation, it is validated that the wave source boundary condition can encompass both the forward and reverse flows, where the numerical results of the full and partial simulations show good agreement in terms of the wave profile, velocity distributions, and measured pressure histories. Thereafter, the full and partial simulations of a tsunami experiment are carried out. The numerical results in the partial simulation agree well with those obtained in the full simulation in terms of the particle distributions, physical quantities, and loading histories (e.g., pressure, acceleration) on the shoreline structure. It is demonstrated that the proposed offline coupling approach using the wave source boundary condition is compatibly utilized with the developed in-house SPH parallel solver, facilitating the detailed FSI simulations within a reduced domain of interest for long-distance tsunami-like phenomena.

Bayesian hybrid-kernel machine-learning-assisted sensitivity analysis and sensitivity-relevant inverse modeling for groundwater DNAPL contamination

Accurate source characterization and transport parameter estimation is important when seeking to predict the spatiotemporal distribution of dense non-aqueous phase liquid (DNAPL) contaminants in groundwater. However, this is a complex multimodal search problem prone to equifinality and premature convergence, which leads to considerable error. To address this, a sensitivity-relevant dynamic swarm intelligence (SRD-SI) algorithm embedded in a homotopy-variation mechanism is proposed in the present study to rationally balance the inversion processes of sensitivity-varied source characteristics and DNAPL transport parameters. In this approach, global optima are progressively approached in conjunction with the homotopy variation of the search space. Furthermore, to avoid computationally expensive numerical model repetition during the sensitivity analysis and inverse iterations, a Bayesian-based optimization framework that combines multiple kernel functions in a kernel extreme learning machine (KELM) model is designed considering the complex site conditions and statistical characteristics of the input variables, thus creating the Bayesian hybrid KELM (BHK-ELM) model for the reliable surrogate modeling of numerical DNAPL-transport simulations. The results show that the BHK-ELM model recognizes and effectively reconstructs the complex input − output mapping of the numerical model by increasing the determination coefficient R2 to 0.9988 while improving the computational efficiency approximately 4500-fold. Because source characteristics and boundary conditions are far more sensitive than transport parameters to the contaminant distribution, conventional inverse modeling methods struggle to accurately identify these transport parameters. In contrast, the proposed inverse modeling system combining sensitivity analysis, swarm intelligence, and homotopy variation is more stable and provides significantly more accurate estimations for all unknown variables. Compared with the traditional SI algorithm, the homotopy-variation SRD-SI reduced the maximum inversion relative error from 46.22 % to 9.53 %, while the mean inversion relative error was reduced from 11.09 % to 3.90 %.

A framework for computing directivities for ultrasonic sources in generally anisotropic, multi-layered media

The knowledge of the directivity of ultrasonic sources is an important element in the design of non-destructive evaluation inspection. The commonly used directivity model of piezoelectric devices is the superposition of out-of-plane force monopoles on the stress-free surface of an isotropic elastic half-space. However, this does not cover a growing range of ultrasonic generation scenarios, as new ultrasonic technologies such as laser ultrasound (LU) emerges, and the need for inspection of materials such as carbon fibre reinforced polymers (CFRPs) rises. A directivity calculation framework based on the reciprocity theorem is developed in the current paper, which accommodates various combinations of source types, material properties, and boundary conditions, for example, a localised heating under a surface layer over an anisotropic half-space. The calculation is applicable to materials with weak anisotropy or wave modes that do not have cuspoidal wavefronts, and was validated with finite element models on key representative cases. The analytical solution for LU excitation in a CFRP sample also shows a good fit to the experimentally measured directivity of the quasi-longitudinal wave mode.

Analysis and experimental research of transient temperature rise characteristics of high-speed cylindrical roller bearing

In this paper, on one hand, the time-varying characteristics of the heat source and thermal boundary conditions of the high-speed spindle system were analyzed considering the thermal-structural coupling effect. And a transient bearing temperature field prediction method combining the thermal network method and finite element method was proposed. Furthermore, the relationships between time step, calculation efficiency and calculation results were analyzed. On the other hand, a online real-time monitoring system of the transient temperature of the cylindrical roller bearing inner ring for maximum speed of 13,000 r/min was designed and implemented using fibre optic sensing technology. Comparing with the conventional static thermal analysis results, it is verified that the simulation method proposed in this paper has higher accuracy. This paper provides a new approach for analysing and testing the thermal characteristics of high-speed spindle system.

Groundwater contaminant source identification considering unknown boundary condition based on an automated machine learning surrogate

Groundwater contamination source identification (GCSI) is a prerequisite for contamination risk evaluation and efficient groundwater contamination remediation programs. The boundary condition generally is set as known variables in previous GCSI studies. However, in many practical cases, the boundary condition is complicated and cannot be estimated accurately in advance. Setting the boundary condition as known variables may seriously deviate from the actual situation and lead to distorted identification results. And the results of GCSI are affected by multiple factors, including contaminant source information, model parameters, boundary condition, etc. Therefore, if the boundary condition is not estimated accurately, other factors will also be estimated inaccurately. This study focuses on the unknown boundary condition and proposed to identify three types of unknown variables (contaminant source information, model parameters and boundary condition) innovatively. When simulation–optimization (S-O) method is applied to GCSI, the huge computational load is usually reduced by building surrogate models. However, when building surrogate models, the researchers need to select the models and optimize the hyperparameters to make the model powerful, which can be a lengthy process. The automated machine learning (AutoML) method was used to build surrogate model, which automates the model selection and hyperparameter optimization in machine learning engineering, largely reducing human operations and saving time. The accuracy of AutoML surrogate model is compared with the surrogate model used in eXtreme Gradient Boosting method (XGBoost), random forest method (RF), extra trees regressor method (ETR) and elasticnet method (EN) respectively, which are automatically selected in AutoML engineering. The results show that the surrogate model constructed by AutoML method has the best accuracy compared with the other four methods. This study provides reliable and strong support for GCSI.

On the equivocal nature of the mass absorption curves

The idea behind the research presented is based upon apparently contradictory experimental results obtained here by means of photoacoustics modalities for the same drug donor/acceptor membrane system, serving as a surrogate for a transdermal delivery system. The first modality allowed for the monitoring of the total amount of mass uptake (m(t)-type data), while the second technique allowed for the quantification of time-dependent concentration distribution within the acceptor membrane (c(x,t)-type data). Despite of a very good agreement between the mt data and the 1st-order uptake fitting model (standard Fickian diffusion with constant source boundary condition), the standard approach failed during the c(x,t) data analysis. The results led to the analysis of the interfacial transfer contribution to the overall mass transfer efficiency, which eventually allowed to question reliability of the mt data analysis for the determination and quantification of the mass transport parameters. A more detailed analysis of the c(x,t) by means of the newly introduced transport rate number parameter revealed, that the mass uptake by the acceptor is almost equally influenced by interfacial and bulk transport processes. The analyses performed were translated into a model-free characteristic times, i.e. parameters independent of the model scheme used.

Electro-Absorption Modulator Thermo-Optical Self-Heating Analysis

In this work a novel methodology is presented for coupled thermo-optical modelling of an electro-absorption modulator (EAM). First, an optical FDTD simulation results in the light absorption map in the EAM waveguide. This absorption is used as a heat source boundary condition in a thermal finite element (FE) simulation in order to calculate the waveguide temperature. The FE model is validated with experimental data. The temperature is averaged over the device length and used as input in a compact, fully coupled thermo-optical model. The model reveals a thermal instability of the device characteristic, which limits the maximum amount of optical power the device can handle. Finally, using the thermal FE model, a self-heating mitigation strategy is proposed. Using process compatible design changes, self-heating can be reduced by up to 40%, greatly increasing the EAM optical power handling capabilities.

Circuitry optimization using genetic programming for the advancement of next generation refrigerants

In this study, a new evolutionary method, which can handle the implementation of genetic operators with unrestrained number and locations of splitting and merging nodes for the optimization of heat exchanger circuitries, is developed. Accordingly, this technique expands the search space of previous optimization studies. To this end, a finned-tube heat exchanger simulator is structured around a bijective mathematical representation of a refrigerant circuitry (the tube–tube adjacency matrix), which is used in combination with traversing algorithms from graph theory to recognize infeasible circuitries and constrain the evolutionary search to coherent and feasible offspring. The performance of three refrigerants, namely R32, R410A, and R454C, commonly used in air-conditioning applications was assessed for the optimized circuitries of a 36-tube evaporator while converging to a given cooling capacity, degree of superheating, and heat source boundary conditions. At a given output capacity and air outlet temperature, larger coefficient-of-performance improvements (up to 9.99% with reference to a common serpentine configuration) were realized for zeotropic refrigerant mixtures, such as R454C, where appropriate matching of the temperature glide with the temperature variation of the air yielded the possibility of further reducing the required compression ratio under the corresponding operating conditions. Hence, it was demonstrated that low-GWP zeotropic mixtures with temperature glide can realize a performance comparable to that of R32 and higher than that of R410A by approaching the Lorenz cycle operation.

Entropy and exergy analysis of coupled radiative heat transfer and heat conduction: A new thermodynamics approach

This study presents a new approach for evaluating the thermodynamic efficiency of a coupled radiative and conductive heat transfer process with an internal heat source. In this approach, the exergy balance equation is derived, and the exergy augmentation rate due to the internal heat source is found to be the sum of the exergy of the system output to the environment surroundings, the exergy stored in the system, and the exergy consumption rate due to the irreversibility of heat transfer in the system. The exergetic efficiency of the coupled heat transfer process is then obtained according to the radiative exergy balance equation. Based on the above theoretical derivation, we carry out a thermodynamic analysis of an example including five coupled heat conduction-radiation transfer systems with a uniform internal heat source and different boundary conditions. The variations in the numerical results are discussed according to the exergy balance principle, and the behaviours of the exergy components over time under transient conditions are analysed. Our numerical results show that increasing the area of the adiabatic boundary and reducing the heat loss of the system should be considered when designing heat-transfer systems (as the exergetic efficiency increases from 0.515 to 0.617). The larger the area of the adiabatic boundary of the system, the stronger the effect of increasing this area on the overall optimisation. When a system has two adiabatic boundaries, a symmetrical arrangement of the adiabatic boundaries is suggested, although this is much less effective than increasing the number of adiabatic boundaries.

Deep convolutional Ritz method: parametric PDE surrogates without labeled data

The parametric surrogate models for partial differential equations (PDEs) are a necessary component for many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool to generate these surrogates when parametric fields are present. CNNs are commonly trained on labeled data based on one-to-one sets of parameter-input and PDE-output fields. Recently, residual-based deep convolutional physics-informed neural network (DCPINN) solvers for parametric PDEs have been proposed to build surrogates without the need for labeled data. These allow for the generation of surrogates without an expensive offline-phase. In this work, we present an alternative formulation termed deep convolutional Ritz method (DCRM) as a parametric PDE solver. The approach is based on the minimization of energy functionals, which lowers the order of the differential operators compared to residual-based methods. Based on studies involving the Poisson equation with a spatially parameterized source term and boundary conditions, we find that CNNs trained on labeled data outperform DCPINNs in convergence speed and generalization abilities. The surrogates generated from the DCRM, however, converge significantly faster than their DCPINN counterparts, and prove to generalize faster and better than the surrogates obtained from both CNNs trained on labeled data and DCPINNs. This hints that the DCRM could make PDE solution surrogates trained without labeled data possibly.

BI-GreenNet: Learning Green’s Functions by Boundary Integral Network

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, in most cases, Green’s function is difficult to compute. The troubles arise in the following threefold. Firstly, compared with the original PDE, the dimension of Green’s function is doubled, making it impossible to be handled by traditional mesh-based methods. Secondly, Green’s function usually contains singularities which increase in the difficulty to get a good approximation. Lastly, the computational domain may be very complex or even unbounded. To override these problems, we develop a new framework for computing Green’s function leveraging the fundamental solution, boundary integral method and neural networks with reasonably high accuracy in this paper. We focus on Green’s function of Poisson and Helmholtz equations in bounded domains, unbounded domains. We also consider Poisson equation and Helmholtz equations in domains with interfaces. Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function. In addition, Green’s function provides the operator from the source term and boundary condition to the PDE solution. We apply Green’s function to solve PDEs with different sources, and obtain reasonably high-precision solutions, which shows the good generalization ability of our method. However, the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs, such as variable coefficient equations, which will be investigated in our future work.

Assessing 1D Hydrodynamic Modeling of Júcar River Behavior in Mancha Oriental Aquifer Domain (SE Spain)

In times of population growth, climate change, and increasing water scarcity around the world, it is important to take an objective look at water, a fundamental resource for life. Hydrodynamic modeling makes possible the research of different aspects of the water cycle and the evaluation of different hydrological and hydrogeological forecasting scenarios in the short and medium terms. The present research offers a more detailed scope at the hydrodynamic processes and their space-time distributions on a UE pilot in the Júcar River Basin, providing a calibrated and validated hydrodynamic model of 121 km river reach for 45 years period (1974–2019) on a daily scale. The obtained information is about discharge and water depths along the Júcar River reach within the hydrogeological boundaries of the Mancha Oriental Aquifer (MOA). The river–aquifer interactions have been represented as dynamic boundary conditions expressed as a difference between observed discharges measured in 3 gauging stations. The obtained calibration error performance evaluations of observed and simulated values cover two periods, according to observed data availability from gauging station 08036 with resulting R2 for both discharges and water depths over 0.96. The model validation results were obtained for a different gauge 08132 and the determination coefficients R2 also perform very well with value of 0.90. The model developed might be useful for decision making in water resources management and can be used to generate simulated time series of water depths, levels, discharges, and velocities in reaches where gauging measurements are not available with a desired space-time resolution (from meter/second to kilometer/month). Estimation of critical discharge value (1.973 m3s−1) for system equilibrium, based on the balance between losing and gaining sub-reaches of the river, is also made with a statistical significance at 95% for hydrologic years 2007–2010, period influenced by restrictions in groundwater withdrawals. The results of the present research are important for the proper and objective management of the scarce water resources on a watershed scale in Júcar River Basin, a complex case study representing semiarid climate, growing anthropogenic pressures, and complex river–aquifer interactions. The used approach of dynamic representation of the river–aquifer interactions as distributed source boundary condition in the one-dimensional hydrodynamic model might be applied in another study case on similar scale.

Blow‐up for a Von Karman equation with nonlinear dissipation, logarithmic source, and acoustic boundary conditions

In this paper, we address a Von Karman equation with nonlinear dissipation, logarithmic source effect, and acoustic boundary conditions. We devote to investigate a blow‐up result of the solutions corresponding to the equation. Owing to the presence of nonlinear dissipation instead of linear dissipation, we have some difficulty in use of the concavity method. Thus, using the energy method, we establish a blow‐up result of the solutions with some positive initial energy.

Effects of copper and titania nanoparticles on MHD 3D rotational flow over an elongating sheet with convective thermal boundary condition

Copper nanoparticles (NPs) are used as highly thermal conductive material as well as alloys. In the same way, titanium dioxide nanoparticles have wide usage in coating, waste water treatment, food industries and space crafts etc. The three dimensional (3D) rotational nanofluid flow and heat transfer over an elongating sheet with thermal radiation, heat source and thermal convective boundary condition is studied. The convective heat transfer currently discussed is the superposition of thermal conduction due to temperature gradient and heat energy transfer by the flowing fluid. This phenomenon has been combined with electrically conducting nanofluid subjected to magnetic field. Further, flow model includes rotation and translation which find applications in chemical and process industries. The boundary conditions on heat transfer involves Biot number (Bi) which is important because most of the problem reported in literature involves Nusselt number which is a relative measure of heat transfer coefficient and thermal conductivity, both related to flowing fluid. The solution involves Runge-Kutta forth order method with MATLAB code. It is found that for metallic nanofluid (Cu) temperature remains positive and for metallic oxide nanofluid (TiO 2) the temperature remains negative, indicating thermal energy generation and absorption due to metal or metallic oxide as nanoparticles.