Discontinuous Problems Research Articles

Hyperparameter selection for physics-informed neural networks (PINNs) – Application to discontinuous heat conduction problems

In recent years, physics-informed neural networks (PINNs) have emerged as an alternative to conventional numerical techniques to solve forward and inverse problems involving partial differential equations (PDEs). Despite its success in problems with smooth solutions, implementing PINNs for problems with discontinuous boundary conditions (BCs) or discontinuous PDE coefficients is a challenge. The accuracy of the predicted solution is contingent upon the selection of appropriate hyperparameters. In this work, we performed hyperparameter optimization of PINNs to find the optimal neural network architecture, number of hidden layers, learning rate, and activation function for heat conduction problems with a discontinuous solution. Our aim was to obtain all the settings that achieve a relative L2 error of 10% or less across all the test cases. Results from five different heat conduction problems show that the optimized hyperparameters produce a mean relative L2 error of 5.60%.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years. Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Network (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neutron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN, PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains from the field of nuclear reactor physics. The methods were compared with the standard finite element method. The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

A general model fitting coleoid cephalopod growth as a function of time and temperature to a single curve

AbstractA novel general model has been developed to fit biomass growth data of coleoid cephalopods as a function of time and temperature from hatching to maximum size in a single curve. This model has been obtained formally from a formulation of product‐integral, solves the discontinuity problem present in two‐phase growth models and describes both asymptotic and non‐asymptotic growth. Within the usual temperature range in which the organism develops, our model describes a non‐asymptotic growth achieving a good fit with the experimental data . In particular, the model provides precise calculation of the time k at which the growth curve inflection occurs, time μ1 at which sexual maturity begins and time μ2 at which the maximum size is reached. When the model was parameterized as a function of temperature, it corroborated different results on Octopus maya fitness, in particular, 25.7°C was obtained as the temperature that generates the highest average growth rate and 21.7°C as the temperature with the highest biomass accumulation over time. This result confirms that the temperature range of 22–26°C is suitable for the growth of O. maya and shows that the developed model can be a useful tool for aquaculture.

Analyzing of continuous and discontinuous contact problems of a functionally graded layer: theory of elasticity and finite element method

Contact mechanics analysis is crucial because such problems often arise in engineering practice. When examining contact mechanics, the material property of the contacting components is a crucially significant aspect. It is more complex to solve the contact mechanics of systems that are composed of materials that do not have a homogenous structure compared to materials that have homogeneous qualities throughout. While many studies on contact problems with homogeneous materials exist, those involving non-homogeneous materials are scarce in the literature. As material technology improves fast, there will be a greater need to solve such problems. In this respect, analytical and finite element method (FEM) solutions of the continuous and discontinuous contact problems of a functionally graded (FG) layer are carried out in this article. The FG layer in the problem rests on a rigid foundation and is pressed with a rigid punch. From the solutions, the contact length, contact stress, initial separation distance, and beginning and ending points of separation were determined, and the results were compared. It has been concluded that the FEM findings are consistent with the analytical results to a satisfactory degree. This study analyzes contact problem using different approaches and accounts for the influence of body force in a contact geometry that has yet to be reported.

Numerical study of soil-rock mixtures with 3D numerical manifold method: Generation of random aggregate structure and discretized model

As the main material composition of many large landslides in China and other countries, the soil-rock mixtures (SRMs) have attracted many researchers to investigate their mechanical and hydraulic characteristics in the past several decades. Although in-door and in-situ tests still play the dominant role in obtaining the material parameters of SRMs, numerical methods are seen as a better choice regarding economy and practicality. As a competitive numerical method, the numerical manifold method (NMM), which can solve continuous and discontinuous problems in a unified framework, is used in the present work to investigate the hydraulic characters of SRMs. Before investigating the hydraulic characters of SRMs with NMM, the aggregate structural model and discretized model of SRMs should be built. To effectively build the aggregate structure model of SRMs with high rock content, a simplified random sequential addition method, which is based on E(A, B) concept and background mesh, is firstly developed. Then, the recently proposed parallel preprocessing strategy of NMM is used to generate the discretized model of SRMs. Finally, an illustrative example regarding the determination of effective permeability coefficient of SRMs is solved with 3DNMM. The proposed numerical model in investigating SRMs deserves further investigation.

Elliptic operators with discontinuous coefficients in meshfree GFDM

AbstractIn phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders of magnitude. These jump discontinuities represent interfaces between the phases, and they emerge naturally during the simulation; thus, their exact location is generally unknown a priori. Our goal is to simulate phase change processes with a meshfree generalized finite difference method in a monolithic model without distinguishing between the different phases. There, the material properties mentioned above appear as coefficients inside elliptic operators in divergence form and the jumps must be treated adequately by the numerical method. We present a numerical method for discretizing elliptic operators with discontinuous coefficients without the need for a domain decomposition or tracking of interfaces. Our method facilitates the construction of diagonally dominant diffusion operators that lead to M‐matrices for the discrete Poisson's equation, and thus, satisfy the discrete maximum principle. We demonstrate the applicability of the new method for the case of smooth diffusivity and discontinuous diffusivity. We show that the method is first‐order accurate for discontinuous diffusion problems and provides second‐order and fourth‐order convergence for continuous diffusion coefficients.

Physics-informed neural network frameworks for crack simulation based on minimized peridynamic potential energy

Physics-informed neural networks (PINNs), which are promising tools for solving nonlinear equations in the absence of labeled data, have been successfully applied for continuum field approximation in fluid mechanics, solid mechanics, thermodynamics, and other scientific and engineering problems. However, it is equally necessary to solve discontinuous problems such as cracking behaviors in structures and materials, which are challenging to describe using partial differential equations in traditional solid mechanics frameworks. In this work, PINNs are established with the constraint of integral–differential equations in bond-based peridynamic to simulate crack initiation and propagation behaviors in quasi-brittle plates. To solve the cracking behaviors, the initial displacements of the plate are determined by minimizing the total potential energy through the optimization of global network parameters. The subsequent displacements during crack initiation or propagation are characterized by transfer learning to accelerate convergence. The simulated crack paths for four cases with or without precracks are consistent with those obtained in existing studies based on experiments or numerical simulations. Moreover, the effects of network regularization and neuron scales on the nonlinear displacement characterization performance are investigated.

EIGENVALUES OF STURM-LIOUVILLE PROBLEMS WITH EIGENPARAMETER DEPENDENT BOUNDARY AND INTERFACE CONDITIONS

In this paper, a regular discontinuous Sturm-Liouville problem which contains eigenparameter in both boundary and interface conditions is investigated. Firstly, a new operator associated with the problem is constructed, and the self-adjointness of the operator in an appropriate Hilbert space is proved. Some properties of eigenvalues are discussed. Finally, the continuity of eigenvalues and eigenfunctions is investigated, and the differential expressions in the sense of ordinary or Fréchet of the eigenvalues concerning the data are given.

A weak Galerkin finite element method for 1D semiconductor device simulation models

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k+1 for the discrete weak derivative space. The optimal order error estimates in a discrete H1 norm and the standard L2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems.

Drag on a sphere in a slow flow of a binary mixture of rarefied gases

The drag force on a sphere in an isothermal binary mixture of rarefied gases is calculated on the basis of the McCormack model for the linearized Boltzmann equation. The diffuse scattering of the gas–surface interaction law is assumed as the boundary condition. The rigid-spheres model of interatomic interaction potential is used as well as ab initio potentials for helium, argon, and krypton gases. The force is obtained in a wide range of the gas rarefaction, which covers the free molecular, transition, and slip regimes of the gas flow. In the free molecular and slip flow regimes, the problem is solved analytically, while in the transition regime, the system of kinetic equations is solved numerically via the discrete velocity method optimized to overcome the problem of discontinuity of the distribution function of molecular velocities on the convex surface. The calculations are carried out for the mixtures helium–argon and helium–krypton at 300 K. In the slip flow regime, the data available in the literature for the viscous slip coefficient of the helium–argon mixture are used, while for the mixture helium–krypton, it is calculated. The influence of the interatomic interaction potential, molar fraction, and ratio of atomic mass of species in the mixture on the drag force is analyzed.

A novel single-step explicit time integration method based on momentum corrector technique for structural dynamic analysis

In this study, a novel single-step explicit time integration method based on momentum corrector technique is proposed. The algorithmic procedure is formulated using cubic B-spline interpolation and Taylor series expansion. The momentum corrector is introduced to enhance solution accuracy of problems, especially for discontinuous load problems. Through the theoretical analysis of local truncation errors, spectral radius, algorithmic damping and period elongation, several parameter cases are obtained to achieve enhanced accuracy and stability. Numerical examples, including a standard single-degree-of-freedom problem, two discontinuous load problems, a seismic response analysis problem, a wave propagation problem and a nonlinear problem, are tested. The results illustrate the new method achieves high level of accuracy for discontinuous load problems, while simultaneously exhibiting significant accuracy improvement when dealing with other problems.

Two-dimensional radial integration displacement discontinuity method with high-order isoparametric elements

To improve the accuracy of Displacement Discontinuity Method and enhance its adaptivity, a general-purpose two-dimensional displacement discontinuity method with both linear and quadratic isoparametric elements has been developed to model engineering discontinuity problems where cracks or cavities are involved. Linear and quadratic isoparametric elements have linear and quadratic distributions of displacement discontinuity, respectively. Both of them belong to the discontinuous element type, in which the geometry shape functions are different from the interpolation shape functions. A new general formulation, based on the boundary integral functions, is given for displacement discontinuity problems with arbitrary boundary shapes. This formulation contains singular and strongly singular integrals which can be evaluated in the sense of Cauchy principal value and Hadamard principal value, respectively. The radial integration technique is applied to perform these singular integrals with sufficiently high accuracy. Five numerical examples, most of which have analytic solutions, are given to illustrate the improved accuracy of the proposed approach. Compared with the constant displacement discontinuity element, the numerical results by using the present isoparametric displacement discontinuity elements show better accuracy.

Accounting for weak interfaces in computing the effective crack energy of heterogeneous materials using the composite voxel technique

AbstractWe establish a computational methodology to incorporate interfaces with lower crack energy than the surrounding phases when computing the effective crack energy of brittle composite materials. Recent homogenization results for free discontinuity problems are directly applicable to the time-discretized Francfort-Marigo model of brittle fracture in the anti-plane shear case, and computational tools were introduced to evaluate the effective crack energy on complex microstructures using FFT-based solvers and a discretization scheme based on a combinatorially consistent grid. However, this approach only accounts for the crack resistance per volume and is insensitive to the crack resistance of the interface which is expected to play a significant role by considerations from materials science. In this work we introduce a remedy exploiting laminate composite voxels. The latter were originally introduced to enhance the accuracy of solutions for elasticity problems on regular voxel grids. We propose an accurate approximation of the effective crack energy of a laminate with weak interface where an explicit solution is available. We incorporate this insight into an efficient algorithmic framework. Finally, we demonstrate the capabilities of our approach on complex microstructures with weak interfaces between different constituents.

Parallel EM optimization using improved pole‐residue‐based neuro‐TF surrogate and isomorphic orthogonal DOE sampling for microwave components design

AbstractDirect electromagnetic (EM) optimization for microwave components design is usually a time‐consuming process. To improve the optimization efficiency, this paper proposes a novel parallel EM optimization technique exploiting improved pole‐residue‐based neuro‐transfer function (neuro‐TF) surrogate and isomorphic orthogonal design of experiment (DOE) sampling strategy. We propose a new sampling method combining isomorphic orthogonal DOE and parallel EM simulations to generate training data for developing the neuro‐TF surrogate. This proposed sampling method can ensure the scattered distribution of data samples in the overall optimization process, thus effectively improving the surrogate accuracy and increasing the optimization speed. We also propose a new pole‐residue tracking technique for order‐changing to solve the discontinuity problem of pole/residues during the neuro‐TF surrogate development. Different from the fixed split ratio in existing pole‐residue tracking technique, the split ratio of poles and residues in the proposed technique is adaptive and determined according to the information of neighboring samples. Therefore, the continuity and smoothness of pole/residues after the splitting are improved, so as the neuro‐TF surrogate accuracy. In addition, the trust region algorithm is exploited during EM optimization to improve the convergence speed. In this way, the proposed EM optimization technique obtains the optimal solution in a shorter time with fewer iterations than the existing techniques. Two examples of EM optimizations of microwave components are used to illustrate the proposed technique.

Effects of mixed corrosion damage on constitutive relation and ductile fracture behavior of butt-welded joints under cyclic loading

The corrosion damage of welded connection regions posed a threat to the reliability and durability of welded steel structures. In this paper, the morphology scanning and cyclic loading tests of butt-welded joints exposed to the salt spray corrosion environment were conducted to investigate the effects of corrosion damage on surface characteristics and hysteresis behavior of butt-welded joints. Three-dimension scanning test results showed that the butt-welded joint under the salt spray environment suffered a kind of mixed corrosion damage. Compared with the thickness loss caused by uniform corrosion, pitting damage tended to cause serious local geometric defects, aggravating the geometric discontinuity problem. For the corroded butt-welded joint, the hysteresis loop area and cycles decreased significantly with the increase of corrosion damage, and the ductile fracture behavior under the cyclic loading gradually disappeared. The numerical comparative analysis of models with two different patterns of corrosion damage showed that the pitting damage caused a severer hysteresis behavior degradation of butt-welded joints compared to the uniform corrosion damage under the same volumetric loss ratio. Particularly, the pitting damage had a significant effect on reducing cyclic deformation and energy dissipation. According to the above experimental and numerical analysis, a modified cyclic plastic fracture criteria considering pitting damage and a nonlinear cyclic constitutive model considering mixed corrosion for the corroded butt-welded joint were proposed respectively to equivalently consider the influence of corrosion geometric damage by weakening the material properties.

A comparative study for determining rock joint normal stiffness with destructive uniaxial compression and nondestructive ultrasonic wave testing

Rock joint a kind of vital discontinuities in natural rock mass. How to accurately and conveniently determine joint normal stiffness is therefore significant in rock mechanics. Here, first, seven existing methods for determining joint normal stiffness were introduced and reviewed, among which Method I (the indirect measurement method), Method II (the direct determination method), Method III (the across-joint strain gauge measurement method) and Method IV (the deformation measuring ring method) are via destructive uniaxial compression testing, while Method V (the best fitting method), Method VI (the rapid evaluation method) and Method VII (the effective modulus method) are through wave propagation principles and nondestructive ultrasonic testing. Subsequently, laboratory tests of intact and jointed sandstone specimens were conducted following the testing requirements and procedures of those seven methods. A comparison among those methods was then performed. The results show that Method I, i.e. the benchmark method, is reliable and stable. Method II has a conceptual drawback, and its accuracy is acceptable at only very low stress levels. Relative errors in the results from Method III are very large. With Method IV, the testing results are sufficiently accurate despite the strict testing environment and complicated testing procedures. The results from Method V are greatly unstable and significantly dependent on the natural frequency of the transducers. The joint normal stiffness determined with Method VI is stable and accurate, although data processing is complex. Method VII could be adopted to determine the joint normal stiffness corresponding to the rock elastic deformation phase only. Consequently, it is suggested that Methods I, IV and VI should be adopted for the measurement of joint normal stiffness. The findings could be helpful in selecting an appropriate method to determine joint normal stiffness and, hence, to better solve discontinuous rock mass problems.

Transient heat conduction modeling in continuous and discontinuous anisotropic materials with the numerical manifold method

Transient heat conduction is a common problem in engineering. However, related studies in anisotropic materials remain relatively scarce for their complexity, compared with isotropic ones. Benefiting from the use of the two cover systems, i.e., the mathematical cover and the physical cover, the numerical manifold method (NMM) provides a unified framework for both continuous and discontinuous analyses. Herein, the NMM is applied to solve transient heat conduction problems with anisotropic materials for continuous and discontinuous cases. Firstly, the fundamental formulas for anisotropic heat conduction are displayed. Then, the NMM approximations for heat conduction in continuous and discontinuous situations are given, and the discrete formulations are derived based on the Galerkin-form weighted residual method and solved by the backward difference scheme. Finally, typical numerical examples are conducted with non-conforming mathematical covers, and the results show that the proposed method can tackle both continuous and discontinuous unsteady anisotropic heat conduction problems with high accuracy and convenience.

A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.

On the Effectiveness of Parameter-Efficient Fine-Tuning

Fine-tuning pre-trained models has been ubiquitously proven to be effective in a wide range of NLP tasks. However, fine-tuning the whole model is parameter inefficient as it always yields an entirely new model for each task. Currently, many research works propose to only fine-tune a small portion of the parameters while keeping most of the parameters shared across different tasks. These methods achieve surprisingly good performance and are shown to be more stable than their corresponding fully fine-tuned counterparts. However, such kind of methods is still not well understood. Some natural questions arise: How does the parameter sparsity lead to promising performance? Why is the model more stable than the fully fine-tuned models? How to choose the tunable parameters? In this paper, we first categorize the existing methods into random approaches, rule-based approaches, and projection-based approaches based on how they choose which parameters to tune. Then, we show that all of the methods are actually sparse fine-tuned models and conduct a novel theoretical analysis of them. We indicate that the sparsity is actually imposing a regularization on the original model by controlling the upper bound of the stability. Such stability leads to better generalization capability which has been empirically observed in a lot of recent research works. Despite the effectiveness of sparsity grounded by our theory, it still remains an open problem of how to choose the tunable parameters. Currently, the random and rule-based methods do not utilize task-specific data information while the projection-based approaches suffer from the projection discontinuity problem. To better choose the tunable parameters, we propose a novel Second-order Approximation Method (SAM) which approximates the original problem with an analytically solvable optimization function. The tunable parameters are determined by directly optimizing the approximation function. We conduct extensive experiments on several tasks. The experimental results show that our proposed SAM model outperforms many strong baseline models and it also verifies our theoretical analysis. The source code of this paper can be obtained from https://github.com/fuzihaofzh/AnalyzeParameterEff\/icientFinetune .

The contact problem of a functionally graded layer under the effect of gravity

AbstractThis paper considers the continuous and discontinuous contact problems of a functionally graded (FG) layer under the effect of gravity. A finite distributed load is applied to the beam from the top and the beam is placed to the top of a rigid foundation. It is assumed that the shear modulus and mass density of the beam vary exponentially whereas Poisson's ratio remains constant. The problem is solved for both continuous and discontinuous cases. In each case, the governing equations and boundary conditions of the problem are reduced to an integral equation by the help of Fourier integral transform. In the continuous contact, the integral equation solved analytically and critical load that causes initial separation between the layer and the rigid foundation, separation distance and contact pressures are investigated for various non‐homogeneity parameters that presents the change in the shear modulus and density and loadings. The singular integral equation in case of discontinuous contact is numerically solved by using corresponding Gauss–Chebyshev quadrature and an iterative scheme. The effect of non‐homogeneity parameters and loading on the separation distance between the layer and the foundation and contact pressures are analyzed. All the results are shown in tables and figures. It is seen that increasing stiffness and decreasing density at the top of the layer reduce maximum pressures but increase separation distances. This study aims to fill the deficiency in the mechanical behavior of the coatings under gravity.