Robust Numerical Integration Research Articles

Robust numerical integration of embedded solids described in boundary representation

Embedded and immersed methods have become essential tools in computational mechanics, as they allow discretizing arbitrarily complex geometries without the need for boundary-fitted meshes. One of their main challenges is the accurate numerical integration of cut elements. Among the various integration schemes developed for this purpose, moment fitting has proven to be a powerful technique that provides highly efficient and accurate integration rules.This publication presents a framework for the robust and efficient numerical integration of embedded solids described by oriented boundary meshes using moment fitting. The developments include an intersection algorithm that aims to drastically accelerate the computation of the necessary moments while achieving high accuracy. A closed surface parameterization of each cut domain is computed to facilitate the direct application of the divergence theorem. The algorithm is subject to a single quality criterion that guarantees the accurate evaluation of boundary integrals. At the same time, it allows to disregard classical mesh criteria, such as high aspect ratios, strongly varying angles, etc., resulting in extremely fast runtimes. In addition, an existing robust flood fill-based element classification scheme is further developed to initiate filling from arbitrary seed elements and to enable parallel execution, increasing its flexibility and efficiency.The successful application of all proposed algorithms to 4948 valid and flawed STLs from the Thingi10K database (Zhou and Jacobson, 2016) demonstrates their extraordinary robustness. In all cases, the wall-clock time scales at most linearly with the number of elements in the background mesh. We show that higher-order quadrature rules on the boundary elements enable efficient computation of the moments via the divergence theorem with near-machine precision. Finally, the presented methodologies are used to perform direct FE analyses on clean and flawed B-Rep models. All proposed algorithms are publicly available in the open-source C++ framework QuESo – Quadrature for Embedded Solids (https://github.com/manuelmessmer/QuESo), where the moment fitting equations are assembled and solved.

Numerical quadrature for the Prandtl Meyer function at high temperature with application for air in nozzles

When the stagnation temperature of the combustion chamber or ambient air increases, the specific heats and their ratio do not remain constant any more, and start to vary with this temperature. The gas remains perfect, except, it will be calorically imperfect and thermally perfect. A new generalized form of the Prandtl Meyer function is developed, by adding the effect of variation of this temperature, lower than the threshold of dissociation. The new relation is presented in the form of integral of a complex analytical function, having an infinite derivative at the critical temperature. A robust numerical integration quadrature is presented in this context. The classical form of the Prandtl Meyer function of a perfect gas becomes a particular case of the developed form. The comparison is made with the perfect gas model for aim to present a limit of its application. The application is for air.

Robust numerical integration on curved polyhedra based on folded decompositions

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the given parametric surfaces. Each pyramid serves as an integration cell with a geometric mapping from a standard parent domain (e.g., a unit cube), where the tensor-product Gauss quadrature is adopted. As no constraint is imposed on the decomposition, certain resulting pyramids may intersect with themselves, and thus their geometric mappings may present negative Jacobian values. We call such cells folded cells and refer to the corresponding decomposition as a folded decomposition. We show that folded cells do not cause any issues in practice as they are only used to numerically compute certain integrals of interest. The same idea can be applied to planar curved polygons as well. We demonstrate both theoretically and numerically that folded cells can retain the same accuracy as the cells with strictly positive Jacobians. On the other hand, folded cells allow for a much easier and much more flexible decomposition for general curved polyhedra, on which one can robustly compute integrals. In the end, we show that folded cells can flexibly and robustly accommodate real-world complex geometries by presenting several examples in the context of immersed isogeometric analysis, where involved sharp features can be well respected in generating integration cells.

A generalized class of strongly stable and dimension-free T-RPMD integrators.

Recent work shows that strong stability and dimensionality freedom are essential for robust numerical integration of thermostatted ring-polymer molecular dynamics (T-RPMD) and path-integral molecular dynamics, without which standard integrators exhibit non-ergodicity and other pathologies [R. Korol et al., J. Chem. Phys. 151, 124103 (2019) and R. Korol et al., J. Chem. Phys. 152, 104102 (2020)]. In particular, the BCOCB scheme, obtained via Cayley modification of the standard BAOAB scheme, features a simple reparametrization of the free ring-polymer sub-step that confers strong stability and dimensionality freedom and has been shown to yield excellent numerical accuracy in condensed-phase systems with large time steps. Here, we introduce a broader class of T-RPMD numerical integrators that exhibit strong stability and dimensionality freedom, irrespective of the Ornstein-Uhlenbeck friction schedule. In addition to considering equilibrium accuracy and time step stability as in previous work, we evaluate the integrators on the basis of their rates of convergence to equilibrium and their efficiency at evaluating equilibrium expectation values. Within the generalized class, we find BCOCB to be superior with respect to accuracy and efficiency for various configuration-dependent observables, although other integrators within the generalized class perform better for velocity-dependent quantities. Extensive numerical evidence indicates that the stated performance guarantees hold for the strongly anharmonic case of liquid water. Both analytical and numerical results indicate that BCOCB excels over other known integrators in terms of accuracy, efficiency, and stability with respect to time step for practical applications.

Simulation of Cyclic Loading on Pipe Elbows Using Advanced Plane-Stress Elastoplasticity Models1

Abstract This work investigates the response of industrial steel pipe elbows subjected to severe cyclic loading (e.g., seismic or shutdown/startup conditions), associated with the development of significant inelastic strain amplitudes of alternate sign, which may lead to low-cycle fatigue. To model this response, three cyclic-plasticity hardening models are employed for the numerical analysis of large-scale experiments on elbows reported elsewhere. The constitutive relations of the material model follow the context of von Mises cyclic elasto-plasticity, and the hardening models are implemented in a user subroutine, developed by the authors, which employs a robust numerical integration scheme, and is inserted in a general-purpose finite element software. The three hardening models are evaluated in terms of their ability to predict the strain range at critical locations, and in particular, strain accumulation over the load cycles, a phenomenon called “ratcheting.” The overall good comparison between numerical and experimental results demonstrates that the proposed numerical methodology can be used for simulating accurately the mechanical response of pipe elbows under severe inelastic repeated loading. Finally, this paper highlights some limitations of conventional hardening rules in simulating multi-axial material ratcheting.

Robust numerical solution for the two parameter solute solvent transport model

Prediction of solute and solvent transport in cells is central to developing and testing cryopreservation protocols. As we show here, however, the models used can be difficult to accurately numerically integrate in some key cases, and thus are a challenge to implement when determining the time dependent cell state during cryoprotectant equilibration and cooling. Exact solution techniques exist for overcoming this problem, but their implementation is also challenging: inversion of a nonlinear function is required that negates much of the utility of the approach. This communication describes a simple approach for more robust numerical integration that can be implemented using any numerical differential equation solver, and can facilitate arbitrarily accurate solutions to transport models without the complication of inversion formulae or complicated numerical integration schemes. Further, a simple relevant example of red blood cell equilibration with 40% glycerol is presented with comments on extending the approach to other settings.

A Nitsche stabilized finite element method for embedded interfaces: Application to fluid-structure interaction and rigid-body contact

In this article, an accurate and robust numerical formulation is presented for the simulation of the fluid-structure interaction in incompressible fluid flow. The incompressible Navier-Stokes equation is discretized with a stabilized finite element framework on the fixed Eulerian grid. Both symmetric and non-symmetric Nitsche's methods are accessed and employed to weakly impose Dirichlet boundary condition along the interface embedded in the element together with the ghost penalty method stabilizing the solution jump across the element edges. An easy-to-implement and robust numerical integration scheme based on a projection approach is proposed. To the author's knowledge, so far, there is no application of a projection-based approach in the field of numerical integration to deal with discontinuities. Therefore, the results presented in this article is considered as a pioneered and novel projection-based approach in the field of numerical integration to deal with embedded discontinuous function. A second-order staggered-partitioned scheme is employed to weakly couple the fluid and structure solvers. A second-order accurate and unconditionally stable time integration scheme is implemented for simulations. Accurate numerical results are obtained in the numerical examples and validation cases, including vortex-induced vibration (VIV), rotation, freely fall and rigid-body contact.

On the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growth

We consider the problem of designing robust numerical integration scheme of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^{\alpha }$, with $\alpha >1$. We propose an (semi-explicit) exponential-Euler scheme for which we obtain a theoretical convergence rate for the weak error. To this aim, we analyze the $C^{1,4}$ regularity of the solution of the associated backward Kolmogorov PDE using its Feynman–Kac representation and the flow derivative of the involved processes. Under some suitable hypotheses on the parameters of the model, we prove a rate of weak convergence of order one for the proposed exponential Euler scheme, and illustrate it with some numerical experiments.

A robust numerical integrator for the short pulse equation near criticality

Abstract The short pulse equation was introduced by Schafer–Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schrodinger equation does not possess, have drawn much attention. In such a region, the solutions can become quite close to a singularity, and thus existing numerical methods cease to work stably, or even if they do, they require high computational cost. In this paper, we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure-preserving methods. The resulting scheme successfully works even when the singularity occurs, and thus gives a highly robust method for resolving near singular solutions of interest. It is confirmed by numerical experiments, and some new insights about derivative blow-up are obtained.

Reflection-mode digital gradient sensing method: measurement accuracy

Surface topography evaluation from measured surface slopes is critical in many engineering applications, including metrology of electronic substrates, optical elements, and small deformation measurement of engineering structures subjected to external thermomechanical stimuli. A recently proposed noncontact full-field optical method called reflection-mode digital gradient sensing (r-DGS) is able to measure small angular deflections of light rays proportional to two orthogonal surface slopes. It has been shown that submicron scale deformations can be detected by processing the slope data using a robust numerical integration scheme called the higher-order finite-difference-based least-squares integration. However, the smallest measurable deformations and the associated accuracy are yet to be determined. In our work, this very issue is addressed by carrying out experiments at temporally different recording frequencies, namely (a) ultrahigh-speed digital photography at 106 frames per second (fps) and (b) slow recording speeds of 101 fps. The results show that r-DGS is able to measure submicron out-of-plane deformations under latter conditions and nanoscale deformations in the former conditions.

Numerical Solution of Fractional Optimal Control

This paper presents a numerical algorithm for solving a class of nonlinear optimal control problems subject to a system of fractional differential equations. We first propose a robust second-order numerical integration scheme for the system. The objective is approximated by the trapezoidal rule. We then apply a gradient-based optimization method to the discretized problem. Formulas for calculating the gradients are derived. Computational results demonstrate that our method is able to generate accurate numerical approximations for problems with multiple states and controls. It is also robust with respect to the fractional orders of derivatives.

Implementation of CPCT based BIEs for 3D elasticity and its application in fracture mechanics

The accurate evaluation of singular boundary integrals is a long-debated issue for the successful implementation of BIEs. In the past decades, various methods have been developed, among which the singularity subtraction technique (SST) has been proved to be a unified method for various orders of singularities. However, the accuracy of original SST is sensitive to element shape due to the near singularity caused by element shape distortion. In this paper, a new conformal polar coordinate transformation (CPCT) for quadrilateral elements is proposed to eliminate the shape effect. Besides, an improved sigmoidal transformation is introduced in the angular direction to assign Gauss points more reasonably. By combination of the two strategies with original SST, a more efficient and robust numerical integration scheme for singular integrals can be obtained. Several numerical examples including crack problems are presented to demonstrate the accuracy and feasibility to coarse meshes for CPCT based BIEs and dual BEM.

In-plane/out-of-plane separated representations of updated Lagrangian descriptions of viscoplastic flow models in plate domains

Abstract A new efficient updated Lagrangian strategy for numerical simulations of material forming processes is presented. The basic ingredient is the tensorial decomposition of the velocity field into a finite sum of in-plane and an out-of-plane components, giving rise to an equivalent computational complexity of some two-dimensional problems and some one-dimensional ones (therefore, much less than the true three-dimensional complexity of the original problem). This is efficiently achieved by using Proper Generalized Decomposition (PGD) techniques, which are here employed in an updated Lagrangian framework for the very first time. This updated Lagrangian nature of the method needs the use of a robust numerical integration technique (in this case, the Stabilized Conforming Nodal Integration has been chosen) for addressing the highly distorted projected meshes. The resulting strategy is of general purpose, although it is especially well suited for addressing models defined in plate or shell (in general, parallelepipedic) domains. The basics of the just-developed method are shown, together with some numerical examples to show the potential of the technique.

Robust adaptive numerical integration of irregular functions with applications to basket and other multi-dimensional exotic options

We improve an adaptive integration algorithm proposed by two of the authors by introducing a new splitting strategy based on a geometrical criterion. This algorithm is tested especially on the pricing of multidimensional vanilla options in the Black–Scholes framework which emphasizes the numerical problems of integrating non-smooth functions. In high dimensions, this new algorithm is used as a control variate after a dimension reduction based on principal component analysis. Numerical tests are performed on the Genz package, on the pricing of basket, put on minimum and digital options in dimensions up to ten.

Efficient Updated-Lagrangian Simulations in Forming Processes

A new efficient updated-Lagrangian strategy for numerical simulations of material forming processes is presented in this work. The basic ingredients are the in-plane-out-of-plane PGD-based decomposition and the use of a robust numerical integration technique (the Stabilized Conforming Nodal Integration). This strategy is of general purpose, although it is especially well suited for plateshape geometries. This paper is devoted to show the feasibility of the technique through some simple numerical examples.

Efficient evaluation of integrals with kernel 1/rχ for quadrilateral elements with irregular shape

In this paper, integrals with kernel 1/rχ are concerned with the following three aspects: a). the near singularity caused by distorted element shape; b). the near singularity derived from the angular direction; c). the singularity/near singularity in the radial direction. A conformal polar coordinate transformation (CPCT) is proposed to eliminate the shape effect of elements, which can keep the shape characteristic of distorted elements, and an improved sigmoidal transformation is introduced to alleviate the near singularity in the angular direction. By combination of the two strategies with existing methods, such as singularity subtraction method and distance transformation method utilized in this paper, an efficient and robust numerical integration approach can be obtained for various orders of singular/nearly singular integrals, and a distorted curved quadrilateral element extracted from a cylinder surface is provided to demonstrate the efficiency and robustness of the proposed method.

Regularization of nonlinear DAEs based on Algebraic Analysis

Differential-algebraic equations (DAEs) arise naturally in many technical and industrial applications. In particular, in automatic modeling using coupling of modularized subcomponents, the coupling of subcomponents is usually described by algebraic constraints which lead to DAEs as model equations. The direct numerical integration of DAEs in general is not feasible due to so-called hidden constraints which are contained in the DAE. The occurrence of hidden constraints leads to difficulties in the numerical integration as instabilities or order reduction can occur. Therefore, it is necessary to regularize or remodel the model equations before a robust numerical integration is possible. We review a remodeling or regularization techniques for general nonlinear DAEs based on an algebraic analysis that has been developed in Kunkel and Mehrmann (2001). This technique can be used to improve the model equations for the numerical treatment.

Regularization of Quasi-Linear Differential-Algebraic Equations

The complete virtual design of dynamical systems, e.g., mechanical systems, electrical circuits, flow problems, or whole production processes, plays a key role in our technological progress. In this context differential-algebraic equations (DAEs) are a very important tool for the analysis and simulation of such dynamical processes. It is well known that the numerical treatment of DAEs is nontrivial in general. The occurrence of hidden constraints (contained in the DAE but only obtainable by differentiating of (parts of) the DAE) impose additional consistency conditions on the initial values and provoke severe difficulties in the direct numerical integration of DAEs as for example drift, instabilities, or convergence problems. Therefore, it is necessary to regularize or remodel the model equations before a robust numerical integration is possible.In this article we present regularization methods for quasi-linear DAEs. In particular, we discuss the projected strangeness-free formulation, the minimally extended formulation, and the overdetermined formulation as regularizations. At the end we give some remarks on the numerical treatment of the presented regularizations.

A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

SUMMARYThis paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley & Sons, Ltd.

Globally Adaptive Control Variate for Robust Numerical Integration

Many methods in computer graphics require the integration of functions on low-to-middle--dimensional spaces. However, no available method can handle all the possible integrands accurately and rapidly. This paper presents a robust numerical integration method, able to handle arbitrary nonsingular scalar or vector-valued functions defined on low-to-middle--dimensional spaces. Our method combines control variate, globally adaptive subdivision and Monte-Carlo estimation to achieve fast and accurate computations of any nonsingular integral. The runtime is linear with respect to standard deviation while standard Monte-Carlo methods are quadratic. We additionally show through numerical tests that our method is extremely stable from a computation time and memory footprint point of view, assessing its robustness. We demonstrate our method on a participating media voxelization application, which requires the computation of several millions integrals for complex media.