A monolithic approach toward coupled electrodynamic–thermomechanical problems with regard to weak formulations

Weak formulations of boundary value problems are the basis of various numerical discretization schemes. They are classically derived applying the method of weighted residuals or a variational principle. For electrodynamical and caloric problems, variational approaches are not straightforwardly obtained from physical principles like in mechanics. Weak formulations of Maxwell’s equations and of energy or charge balances thus are frequently derived from the method of weighted residuals or tailored variational approaches. Related formulations of multiphysical problems, combining mechanical balance equations and the axioms of electrodynamics with those of heat conduction, however, raise the additional issue of lacking consistency of physical units, since fluxes of charge and heat intrinsically involve time rates and temperature is only included in the heat balance. In this paper, an energy-based approach toward combined electrodynamic–thermomechanical problems is presented within a classical framework, merging Hamilton’s and Jourdain’s variational principles, originally established in analytical mechanics, to obtain an appropriate basis for a multiphysical formulation. Complementing the Lagrange function by additional potentials of heat flux and electric current and appropriately defining generalized virtual powers of external fields including dissipative processes, a consistent formulation is obtained for the four-field problem and compared to a weighted residuals approach.

Application of exponential functions in weighted residuals method in structural mechanics. Part 3: infinite cylindrical shell under concentrated forces

Solution for cylindrical shell under concentrated force is a fundamental problem which allow to consider many other cases of loading and geometries. Existing solutions were based on simplified assumptions, and the ranges of accuracy of them still remains unknown. The common idea is the expansion of them into Fourier series with respect to circumferential coordinate. This reduces the problem to 8th order even differential equation as to axial coordinate. Yet the finding of relevant 8 eigenfunctions and exact relation of 8 constant of integrations with boundary conditions are still beyond the possibilities of analytical treatment. In this paper we apply the decaying exponential functions in Galerkin-like version of weighted residual method to above-mentioned 8th order equation. So, we construct the sets of basic functions each to satisfy boundary conditions as well as axial and circumferential equilibrium equations. The latter gives interdependencies between the coefficients of circumferential and axial displacements with the radial ones. As to radial equilibrium, it is satisfied only approximately by minimizations of residuals. In similar way we developed technique for application of Navier like version of WRM. The results and peculiarities of WRM application are discussed in details for cos2j concentrated loading, which methodologically is the most complicated case, because it embraces the longest distance over the cylinder. The solution for it clearly exhibits two types of behaviors – long-wave and short-wave ones, the analytical technique of treatment of them was developed by first author elsewhere, and here was successfully compared. This example demonstrates the superior accuracy of two semi analytical WRM methods. It was shown that Navier method while being simpler in realization still requires much more (at least by two orders) terms than exponential functions.

An improved data-free surrogate model for solving partial differential equations using deep neural networks

Partial differential equations (PDEs) are ubiquitous in natural science and engineering problems. Traditional discrete methods for solving PDEs are usually time-consuming and labor-intensive due to the need for tedious mesh generation and numerical iterations. Recently, deep neural networks have shown new promise in cost-effective surrogate modeling because of their universal function approximation abilities. In this paper, we borrow the idea from physics-informed neural networks (PINNs) and propose an improved data-free surrogate model, DFS-Net. Specifically, we devise an attention-based neural structure containing a weighting mechanism to alleviate the problem of unstable or inaccurate predictions by PINNs. The proposed DFS-Net takes expanded spatial and temporal coordinates as the input and directly outputs the observables (quantities of interest). It approximates the PDE solution by minimizing the weighted residuals of the governing equations and data-fit terms, where no simulation or measured data are needed. The experimental results demonstrate that DFS-Net offers a good trade-off between accuracy and efficiency. It outperforms the widely used surrogate models in terms of prediction performance on different numerical benchmarks, including the Helmholtz, Klein–Gordon, and Navier–Stokes equations.

THERMAL STRESSES FOR FUNCTIONALLY GRADED MATERIALS (FGMS) SUBJECT TO HEAT FLUX

In this paper, the thermal stress due to heat flux at the far field is derived for an infinitely extended elastic medium which contains a spherical inclusion made of functionally graded materials (FGMs). The 3-D heat conduction equation subject to uniform heat flux at the far field is solved analytically to derive the temperature distribution. Based on the temperature solution, the thermal stress field due to heat flux is obtained by solving a set of two ordinary differential equations using the method of weighted residuals. Unlike the two-phase homogeneous medium, the von Mises stress distribution is continuous at the interface of the FGM-matrix medium.

External Evaluation of Population Pharmacokinetic Models and Bayes-Based Dosing of Infliximab

Despite the well-demonstrated efficacy of infliximab in inflammatory diseases, treatment failure remains frequent. Dose adjustment using Bayesian methods has shown in silico its interest in achieving target plasma concentrations. However, most of the published models have not been fully validated in accordance with the recommendations. This study aimed to submit these models to an external evaluation and verify their predictive capabilities. Eight models were selected for external evaluation, carried out on an independent database (409 concentrations from 157 patients). Each model was evaluated based on the following parameters: goodness-of-fit (comparison of predictions to observations), residual error model (population weighted residuals (PWRES), individual weighted residuals (IWRES), and normalized prediction distribution errors (NPDE)), and predictive performances (prediction-corrected visual predictive checks (pcVPC) and Bayesian simulations). The performances observed during this external evaluation varied greatly from one model to another. The eight evaluated models showed a significant bias in population predictions (from −7.19 to 7.38 mg/L). Individual predictions showed acceptable bias and precision for six of the eight models (mean error of −0.74 to −0.29 mg/L and mean percent error of −16.6 to −0.4%). Analysis of NPDE and pcVPC confirmed these results and revealed a problem with the inclusion of several covariates (weight, concomitant immunomodulatory treatment, presence of anti-drug antibodies). This external evaluation showed satisfactory results for some models, notably models A and B, and highlighted several prospects for improving the pharmacokinetic models of infliximab for clinical-biological application.

Application of exponential functions in weighted residuals method in structural mechanics. Part II: static and vibration analysis of rectangular plate

The paper is continuation of our efforts on application of the properly constructed sets of exponential functions as the trial (basic) functions in weighted residuals method, WRM, on example of classical tasks of structural mechanics. The purpose of this paper is justification of new method’s efficiency as opposed to getting new results. So, static deformation and free vibration of isotropic thin – walled plate are considered here. Another peculiarity of paper is choice of weight (test) functions, where three options are investigated: it is the same as trial one (Galerkin method); it is taken as results of application of differential operator to trial function (least square method); it equals to the second derivative of trial function with respect to both x and y coordinate (moment method). Solution is considered as product of two independent sets of functions with respect to x or y coordinates. Each set is the combination of five consequent exponential functions, where coefficient at first function is equal to one, and four other coefficients are to satisfy two boundary conditions at each opposite boundary. The only arbitrary value in this method is the scaling factor at exponents, the reasonable range of which was carefully investigated and was shown to have a negligible impact on results. Static deformation was investigated on example of simple supported plate when outer loading is either symmetrical and concentrated near the center or is shifted to any corner point. It was demonstrated that results converge to correct solution much quickly than in classical Navier method, while moment method seems to be a best choice. Then method was applied to free vibration analysis, and again the accuracy of results on frequencies and mode shape were excellent even at small number of terms. At last the vibration of relatively complicated case of clamped – clamped plate was analyzed and very encouraged results as to efficiency and accuracy were achieved.