Numerical Integration Scheme Research Articles

On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose several alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed numerical integration scheme. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks.

Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law

In [7], we introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.

A cell-based smoothed finite element method for finite elasticity

In this study, we present a displacement based polygonal finite element method for compressible and nearly-incompressible elastic solids undergoing large deformations in two dimensions. This is achieved by projecting the dilatation strain onto the linear approximation space, within the framework of volume averaged nodal projection method. To reduce the numerical integration burden over polytopes, a linear strain smoothing technique is employed to compute the terms in the bilinear/linear form. The salient features of the proposed framework are: (a) does not require derivatives of shape functions and complex numerical integration scheme to compute the bilinear and linear form and (b) volumetric locking is alleviated by adopting the volume averaged nodal projection technique. The efficacy, convergence properties and accuracy of the proposed framework is demonstrated through four standard benchmark problems.

AN EFFICIENT NUMERICAL SCHEME FOR STATIC TORQUE PROFILING OF SWITCHED RELUCTANCE MACHINE WITH EXPERIMENTAL VALIDATIO

In AC and DC drives, the use of switched reluctance machine (SRM) is becoming popular as it has a few preferences over AC and DC drives in a simple and robust construction without brushes, low inertia, and high torque to weight ratio, without rotor windings, simple circuit power converter, etc. SRM is widely used in variable speed and servo drives. Because of the double saliency structure and the high nonlinearity of magnetic material, it is difficult to represent the flux-linkage and static torque characteristic of the SRM. This work promotes the use of an improved numerical integration scheme for the static characteristics of SRM. The static torque function that depends on the rotor position and the phase current of the flux linkage features family (for different rotor positions) is improved in the proposed work. Firstly, we use an experimental setup for the electromagnetic characteristic of SRM. Then, we use the improved scheme to develop an efficient mathematical model for static characteristics and finally simulate the static characteristic of SRM through MATLAB code. In the last step, we compare the performance of the proposed integrator model with an existing approach for better efficiency of the static characteristic of SRM with experimental validation.

Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas

In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler–Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

Stable linear multistep methods with off-step points for the solution of ordinary differential equations

Of recent, stability has become an important concept and a qualitative property in any numerical integration scheme. In this work, we propose two stable linear multistep methods with off-step points for the numerical integration of ordinary differential equations whose development is collocation and interpolation based. The boundary locus techniques show that the proposed schemes are zero-stable, A-stable and -stable for some step number and are found suitable for stiff differential equations. Numerical results obtained compare favourably with some existing methods in literature.

A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state.

A SEIR Epidemic Model of Whooping Cough‐Like Infections and Its Dynamically Consistent Approximation

Whooping cough is a highly transmitted disease around the world. According to the World Health Organization (WHO), 0.15 million cases had reported globally in 2018. Most of the Asian and African states are infected regions. Through the study, we investigated the whole population into the four classes susceptible (S), exposed (E), infected (I), and vaccinated or recovered (R). The transmission dynamics of whooping cough disease are studied analytically and numerically. Analytical analyses are positivity, boundedness, reproduction number, equilibria, and local and global stabilities. In numerical analysis, we developed an implicit numerical integration scheme consistent with the biological problem’s properties. The analysis of the implicit method for the said model is dynamically consistent, positive, and bounded. Furthermore, an implicit numerical integration scheme is suitable for studying a particular epidemic model such as the whooping cough disease.

Machine Learning Enhanced Boundary Element Method: Prediction of Gaussian Quadrature Points

This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods. A model based on the neural network multi-classification algorithm is constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy. The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected. The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model, and the accuracy of the model is about 90%. Finally, by incorporating the predicted Gaussian quadrature points into the boundary element analysis, we find that the numerical solution and the analytical solution are in good agreement, which verifies the robustness of the proposed method.

The dynamics of bouncing, partially coalescing, liquid metal droplets in a viscous medium

Planar partial coalescence is a phenomenon in which a droplet at a free surface or interface between two fluids coalesces into the plane surface producing a smaller droplet rather than coalescing completely. This smaller, ‘daughter’ droplet will be driven towards the interface by gravity and capillary forces resulting in a cascade effect of progressively small daughter droplets until the Ohnesorge Number approaches $\sim$ 1 and the cascade terminates with a full coalescence event. This paper utilizes a room temperature liquid metal alloy composed of gallium, indium and tin to study partial coalescence in a viscous quiescent medium and observed bouncing of the coalescing droplets on the interface. We observed the event using high speed videography measuring effects such as the droplet to daughter droplet ratio, droplet velocities, droplet bounce heights and coefficients of restitution for the bouncing event. An existing model (Honey & Kavehpour, Phys. Rev. E, vol. 73, 2006) from our group was used, validated and expanded upon to include buoyancy effects to estimate the initial velocity of the droplet and we developed two new models for the droplet travel and maximum bounce height. The first utilizes the Stokes model for drag to moderate success while the second utilizes a model from Beard & Pruppacher (J. Atmos. Sci., vol. 26, 1969, pp. 1066–1072) and a fourth-order Runge–Kutta numerical integration scheme to predict the droplet velocity and position as functions of time. Additionally the coefficient of restitution was determined from the model using a shooting method technique in tandem with measured data to find a coefficient of restitution value of $A = 0.27 \pm 0.06$ . This ‘bouncing drop’ phenomenon continues in a quiescent viscous fluid to the sub-micron scale and was facilitated by the material properties of the liquid metal including the high density, moderate viscosity and particularly high interfacial tension.

Two-step Brillouin zone sampling for efficient computation of electron dynamics in solids

We develop a numerical Brillouin-zone integration scheme for real-time propagation of electronic systems with time-dependent density functional theory. This scheme is based on the decomposition of a large simulation into a set of small independent simulations. The performance of the decomposition scheme is examined in both linear and nonlinear regimes by computing the linear optical properties of bulk silicon and high-order harmonic generation. The decomposition of a large simulation into a set of independent simulations can improve the efficiency of parallel computation by reducing communication and synchronization overhead and enhancing the portability of simulations across a relatively small cluster machine.

A novel strain-based finite element family for mesh-independent analysis of the tensile failure of reinforced concrete bars

The paper presents a novel family of strain-based beam finite elements (FE) for analysis of tensile failure of a reinforced concrete bar (RC bar), with results of the analysis being independent of the applied FE mesh. The composite bar consists of a continuous longitudinal ductile reinforcing bar(s) surrounded by brittle concrete cover, which are considered separately in the model. Longitudinal slip at the contact between the concrete cover and reinforcing bars is allowed, while their relative displacements perpendicular to the axis of the RC bar are prevented. Cracks in concrete cover occur when tensile stress in concrete exceeds its tensile strength. Propagation of partially connected crack, that is, softening of the material at the crack, is described through constitutive law in form of nonlinear relationship between stresses in concrete at the crack and the width of the crack. Each separate crack is considered discretely as a discontinuity in geometry of the element. In the analysis of cracking of concrete, it is commonly assumed that the discrete crack can occur at the nodes of FE only. However, this assumption leads to dependence of the analysis results on the employed FE mesh. The presented family of FE enables occurrence of the crack anywhere along the FE. In order to achieve this, the discrete crack is excluded from equations of FE and additional boundary conditions are introduced at the discontinuity. This approach ensures that the location of the cracks, their number and their propagation are independent of the applied FE mesh. Advantages of the novel family of FE are thoroughly presented in a parametric study which investigates influence of number of FE as well as influence of degrees of interpolation and integration on the cracking of RC bar under tensile loading. Experimental results of tensile tests on the analysed bar are available in literature. It can be concluded that the results obtained with the minimal possible number of novel FE and sufficiently high degree of numerical integration scheme, applied for solving integrals in equations of FE, are considerably more accurate than the results of previous analyses with model of discrete crack at the nodes of FE only.

Seismic response analysis of complicated soil-structure interaction using inelasticity-separated scaled boundary finite element method

Inelastic seismic response analysis of large infrastructure needs to consider the soil-structure dynamic interaction with a reasonable absorbing boundary condition and generally requires refined simulation, which is an extremely complicated and time-consuming process. Because these structures are generally irregular, the use of a conventional finite element mesh for refined simulation makes the process of modelling pretreatment more complicated, and the corresponding computation scale is usually large. Although the development of computer hardware has greatly improved the computational efficiency of nonlinear problems, studies related to an efficient solving algorithm and appropriate element formulation that can achieve refined simulation, flexible mesh generation, and high precision for large-scale complex structures are still the focus of civil engineers. Because the material nonlinearity of a structure generally occurs in local regions, various emerging algorithms have been developed to accelerate the computation of inelastic response analyses based on various methods such as the reanalysis method and the inelasticity-separated finite element method (IS-FEM). The scaled boundary finite element method (SBFEM), which is a semi-analytical method, is known for its flexible meshing capability and high precision. The idea of inelasticity-separated is introduced into the SBFEM, and the viscous-spring artificial boundary is combined with a novel inelasticity-separated S-element formulation to propose a dynamic solution scheme that can solve the problems associated with element meshing and the nonlinearity of models with complicated soil-structure dynamic interaction. To implement the semi-analytical S-element for use in inelasticity-separated method, the Gaussian numerical integration scheme is performed for each subelement. The proposed dynamic solution scheme for soil-structure interaction is basically the same as that of a static problem and can also be solved by the efficient Woodbury approximation approach because a more inelastic strain interpolation point layout in the S-element leads to larger inelastic degrees of freedom (IDOFs). In this paper, not only the mesh flexibility and high precision of the SBFEM is retained but also the nonlinear solving efficiency is greatly improved without decreasing precision. Numerical examples demonstrate the validity and efficiency of the proposed inelasticity-separated S-element formulation and dynamic solution scheme.

The mixed Deep Energy Method for resolving concentration features in finite strain hyperelasticity

The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations, i.e. domains with holes, notches, etc. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads, but additionally offers unique capabilities for the solution of inverse problems and parameter estimation in the context of hyperelasticity.

Stochastic dynamical systems developed on Riemannian manifolds

We propose a method for developing the flows of stochastic dynamical systems, posed as Ito’s stochastic differential equations, on a Riemannian manifold identified through a suitably constructed metric. The framework used for the stochastic development, viz. an orthonormal frame bundle that relates a vector on the tangent space of the manifold to its counterpart in the Euclidean space of the same dimension, is the same as that used for developing a standard Brownian motion on the manifold. Mainly drawing upon some aspects of the energetics so as to constrain the flow according to any known or prescribed conditions, we show how to expediently arrive at a suitable metric, thus briefly demonstrating the application of the method to a broad range of problems of general scientific interest. These include simulations of Brownian dynamics trapped in a potential well, a numerical integration scheme that reproduces the linear increase in the mean energy of conservative dynamical systems under additive noise and non-convex optimization. The simplicity of the method and the sharp contrast in its performance vis-á-vis the correspondent Euclidean schemes in our numerical work provide a compelling evidence to its potential, especially in the context of numerical schemes for systems with the ready availability of an energy functional, e.g. those in nonlinear elasticity.

AutoMat: automatic differentiation for generalized standard materials on GPUs

We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness and the applicability of AutoMat by integrating it into the FFT-based homogenization scheme of Moulinec and Suquet and discuss the benefits of using AutoMat with respect to runtime and solution accuracy for an elasto-viscoplastic example.

Comparison of implicit and explicit numerical integration schemes for a bounding surface soil model without elastic range

Standard integration schemes for rate constitutive equations were designed within the classical theory of plasticity. Consequently, they rely on the assumption that a yield criterion defines a range of purely elastic material behaviour. Many constitutive models for non-cohesive soils discard that assumption. They account for inelastic deformations without employing a yield criterion. This simplifies the formulation of the models but raises questions concerning their numerical implementation. In particular, it is not fully clear which algorithmic method is most appropriate for the numerical integration of constitutive models without elastic range. To investigate this, two different stress point algorithms for a critical state bounding surface model for sands were developed and implemented. The explicit method employs substepping to automatically control the local error. The implicit method updates stresses and state variables through a local Newton iteration, the Jacobian of which is computed by numerical differentiation. The two algorithms were compared by means of calculations at integration point level as well as with respect to a boundary value problem. The results show that for a given level of accuracy, the explicit update procedure is significantly more efficient than the implicit one. This holds regardless of initial state parameters and input strain increment magnitudes.

A comprehensive implicit substepping integration scheme for multisurface plasticity

AbstractA complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state‐update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multisurface plasticity models. This algorithm is based on the backward Euler method aiming at accuracy and stability, and on the Newton–Raphson method to solve the unconstrained optimization problem. In this scenario, a line search strategy is adopted to improve the convergence characteristics of the algorithm. The golden section method, an exact line search, is considered. Also, a substepping scheme is implemented to provide additional robustness to the state‐update procedure. Therefore, this work contributes to computational plasticity presenting an adaptive substep size scheme and a consistent tangent modulus according to the substepping technique. Finally, some numerical problems are evaluated using the proposed algorithm. Single‐surface and novel multisurface plasticity models are employed in these analyses. The results testify how the line search and substepping strategies can improve the robustness of the nonlinear analysis.

Wireless, Accumulation Mode Dosimeters for Monitoring Pulsed and Non-Pulsed Germicidal Lamps

Mitigating the spread of infectious diseases such as the one associated with the COVID-19 pandemic demands simple and effective disinfection techniques. Ultraviolet germicidal irradiation (UVGI) is one such method, in common use for decontamination of hospital rooms. Practical technologies designed to monitor UVGI ensure the delivery of sufficient doses for germicidal efficacy. Existing UVGI dosimeters rely on intermittent measurements of intensity as the basis for a numerical integration scheme that approximates dose. Traditional devices are ineffective, particularly with pulsed UVGI lamps that emit pulses of light with durations shorter than the interval of measurement. Here, we present a compact, accumulation mode dosimeter (AMD) that detects continuously, as opposed to intermittently, at single or multiple UVGI wavelengths. The AMD utilizes an array of photodiodes and supercapacitors to passively transduce and capture photocurrent generated by UVGI without the use of external power. The accumulated voltage across the supercapacitors then serves as a measure of UVGI dose. A key result is that sampling intervals of AMD do not constrain measurement accuracy. When implemented with a wireless transponder, AMD supports a light-adaptive sampling scheme designed to adjust the sampling interval to the intensity and period of UVGI exposure. Compared to time-based sampling schemes adopted by conventional sensors, light-adaptive approaches autonomously optimize battery life by minimizing current consumption during periods of low or no UVGI. Benchtop studies of the use of this technology with pulsed Xenon lamps (pulse ~5 ms) sampled at long intervals (>1s) highlight the key features of operation. Demonstration of AMD during UVGI of E. Coli cultures represents an example in dose dependent effects on disinfection.

Probabilistic characterization of the effect of transient stochastic loads on the fatigue-crack nucleation time

The rainflow counting algorithm for material fatigue is both simple to implement and extraordinarily successful for predicting material failure times. However, it neglects memory effects and time-ordering dependence, and therefore runs into difficulties dealing with highly intermittent or transient stochastic loads with heavy tailed distributions. Such loads appear frequently in a wide range of applications in ocean and mechanical engineering, such as wind turbines and offshore structures. In this work we employ the Serebrinsky–Ortiz cohesive envelope model for material fatigue to characterize the effects of load intermittency on the fatigue-crack nucleation time. We first formulate efficient numerical integration schemes, which allow for the direct characterization of the fatigue life in terms of any given load time-series. Subsequently, we consider the case of stochastic intermittent loads with given statistical characteristics. To overcome the need for expensive Monte-Carlo simulations, we formulate the fatigue life as an up-crossing problem of the coherent envelope. Assuming statistical independence for the large intermittent spikes and using probabilistic arguments we derive closed expressions for the up-crossing properties of the coherent envelope and obtain analytical approximations for the probability mass function of the failure time. The analytical expressions are derived directly in terms of the probability density function of the load, as well as the coherent envelope. We examine the accuracy of the analytical approximations and compare the predicted failure time with the standard rainflow algorithm for various loads. Finally, we use the analytical expressions to examine the robustness of the derived probability distribution for the failure time with respect to the coherent envelope geometrical properties.