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At least 847 records · Page 47

A new second-order integration algorithm for simulating mechanical dynamic systems

A new integration algorithm which has the simplicity of Euler integration but exhibits second-order accuracy is described. In fixed-step numerical integration of differential equations for mechanical dynamic systems the method represents displacement and acceleration variables at integer step times and velocity variables at half-integer step times. Asymptotic accuracy of the algorithm is twice that of trapezoidal integration and ten times that of second-order Adams-Bashforth integration. The algorithm is also compatible with real-time inputs when used for a real-time simulation. It can be used to produce simulation outputs at double the integration frame rate, i.e., at both half-integer and integer frame times, even though it requires only one evaluation of state-variable derivatives per integration step. The new algorithm is shown to be especially effective in the simulation of lightly-damped structural modes. Both time-domain and frequency-domain accuracy comparisons with traditional integration methods are presented. Stability of the new algorithm is also examined.

Howe, R. M.↗

PetIGA: A framework for high-performance isogeometric analysis

We present PetIGA, a code framework to approximate the solution of partial differential equations using isogeometric analysis. PetIGA can be used to assemble matrices and vectors which come from a Galerkin weak form, discretized with Non-Uniform Rational B-spline basis functions. We base our framework on PETSc, a high-performance library for the scalable solution of partial differential equations, which simplifies the development of large-scale scientific codes, provides a rich environment for prototyping, and separates parallelism from algorithm choice. We describe the implementation of PetIGA, and exemplify its use by solving a model nonlinear problem. To illustrate the robustness and flexibility of PetIGA, we solve some challenging nonlinear partial differential equations that include problems in both solid and fluid mechanics. Lastly, we show strong scaling results on up to 4096 cores, which confirm the suitability of PetIGA for large scale simulations.

97 MATHEMATICS AND COMPUTING↗

Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems

This paper focuses on a time-varying constrained nonconvex optimization problem, and considers the synthesis and analysis of online regularized primal-dual gradient methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed regularized primal-dual gradient method is implemented in a running fashion, in the sense that the underlying optimization problem changes during the execution of the algorithms. In order to study its performance, we first derive its continuous-time limit as a system of differential inclusions. We then study sufficient conditions for tracking a KKT trajectory, and also derive asymptotic bounds for the tracking error (as a function of the time-variability of a KKT trajectory). Further, we provide a set of sufficient conditions for the KKT trajectories not to bifurcate or merge, and also investigate the optimal choice of the parameters of the algorithm. Illustrative numerical results for a time-varying nonconvex problem are provided.

differential inclusion↗

A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations

Differential equations (DEs) serve as fundamental tools in mathematical modeling across scientific disciplines, yet classical numerical solvers face limitations with large-scale or computationally intensive problems. This study explores a quantum-inspired approach to solving DEs, combining quantum- inspired techniques with classical methods. It focuses on fixed- point arithmetic on quantum circuits, utilizing basic quantum gates to manipulate DE solutions. We expand upon the techniques introduced by Zanger et al. [Quantum, 5, 502 (2021)] by offering a precise computation for a fixed-point signed multiplication scheme, while also presenting a quantum circuit capable of executing the fixed-point division algorithm. We demonstrate the feasibility of our approach through the simulation of a linear Ordinary Differential Equation (ODE), where initial conditions and parameters are encoded into quantum circuits using fixed- point representation. By executing sequences of quantum gates mimicking numerical integration steps, we obtain approximate solutions to the ODE with specified fixed-point precision.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations

Differential equations (DEs) serve as fundamental tools in mathematical modeling across scientific disciplines, yet classical numerical solvers face limitations with large-scale or computationally intensive problems. This study explores a quantum-inspired approach to solving DEs, combining quantum-inspired techniques with classical methods. It focuses on fixed-point arithmetic on quantum circuits, utilizing basic quantum gates to manipulate DE solutions. We expand upon the techniques introduced by Zanger et al. [Quantum, 5, 502 (2021)] by offering a precise computation for a fixed-point signed multiplication scheme, while also presenting a quantum circuit capable of executing the fixed-point division algorithm. We demonstrate the feasibility of our approach through the simulation of a linear Ordinary Differential Equation (ODE), where initial conditions and parameters are encoded into quantum circuits using fixed-point representation. By executing sequences of quantum gates mimicking numerical integration steps, we obtain approximate solutions to the ODE with specified fixed-point precision.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

An overlapping domain decomposition method for large-scale problems

Discretization of dynamical models defined through partial differential equations leads to large-scale systems. Time-depending condition involves an iterative integration of such kind of systems. In this paper, a novel technique based on overlapped domain decomposition, without preconditioner and scalable, is presented. Due to the domain decomposition, subproblems are solved in parallel without communications, cutting off the computation time and optimizing the computational cost. This direct method describes an optimized parallel strategy to solve the initial problem, providing the exact solution, up to rounding errors. Moreover, it takes into account both physical nature of the problem and deriving numerical properties of the system. It is highly recommended in case of band matrices and a long time interval because of an increasing gain in terms of performance and computational cost with the number of integrations. A deep analysis of the computational cost concludes the paper.

97 MATHEMATICS AND COMPUTING↗

A method for solution of the Euler-Bernoulli beam equation in flexible-link robotic systems

An efficient numerical method for solving the partial differential equation (PDE) governing the flexible manipulator control dynamics is presented. A finite-dimensional model of the equation is obtained through discretization in both time and space coordinates by using finite-difference approximations to the PDE. An expert program written in the Macsyma symbolic language is utilized in order to embed the boundary conditions into the program, accounting for a mass carried at the tip of the manipulator. The advantages of the proposed algorithm are many, including the ability to (1) include any distributed actuation term in the partial differential equation, (2) provide distributed sensing of the beam displacement, (3) easily modify the boundary conditions through an expert program, and (4) modify the structure for running under a multiprocessor environment.

Tzes, Anthony P.↗

Mueller-matrix mapping of biological tissues in differential diagnosis of optical anisotropy mechanisms of protein networks

We report a model of Mueller-matrix description of optical anisotropy of protein networks in biological tissues with allowance for the linear birefringence and dichroism. The model is used to construct the reconstruction algorithms of coordinate distributions of phase shifts and the linear dichroism coefficient. In the statistical analysis of such distributions, we have found the objective criteria of differentiation between benign and malignant tissues of the female reproductive system. From the standpoint of evidence-based medicine, we have determined the operating characteristics (sensitivity, specificity and accuracy) of the Mueller-matrix reconstruction method of optical anisotropy parameters and demonstrated its effectiveness in the differentiation of benign and malignant tumours. (laser applications and other topics in quantum electronics)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Direct and implicit optical matrix-vector algorithms

New direct and implicit algorithms for optical matrix-vector and systolic array processors are considered. Direct rather than indirect algorithms to solve linear systems and implicit rather than explicit solutions to solve second-order partial differential equations are discussed. In many cases, such approaches more properly utilize the advantageous features of optical systolic array processors. The matrix-decomposition operation (rather than solution of the simplified matrix-vector equation that results) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system. The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution. Extensions to eigenvalue computation and formation of matrices of special structure are also noted.

Casasent, D.↗

Decomposition of Feynman integrals by multivariate intersection numbers

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

KIVA-Update June 2012

Development of a fractional step, a Predictor-Corrector Split (PCS), or what is often known as a projection method combining hp-adaptive system in a Finite Element Method (FEM) for combustion modeling has been achieved. This model will advance the accuracy and range of applicability of the KIVA combustion model and software used typically for internal combustion engine modeling. This abstract describes a PCS hp-adaptive FEM model for turbulent reactive flow spanning all velocity regimes and fluids that is being developed for the new KIVA combustion algorithm, particularly for internal combustion engines. The method and general solver is applicable to Newtonian and non- Newtonian fluids and also for incompressible solids, porous media, solidification modeling, and fluid structure interaction problems. The fuel injection and injector modeling could easily benefit from the capability of solving the fluid structure interaction problem in an injector, helping to understand cycle to cycle variation and cavitation. This is just one example where the new algorithm differs from the old, in addition to handling Conjugate Heat Transfer (CHT), although there a numerous features that makes the new system more robust and accurate. In these ways, the PCS hp-adaptive algorithm does not compete with commercial software packages, those often used in conjunction with the currently distributed KIVA codes for engine combustion modeling. In addition, choosing a local ALE method on immersed moving parts represented by overset grid that is 2nd order spatially accurate, allows for easy grid generation from CAD to fluid grid while also provide for robustness in handling any possible moving parts configuration without any code modifications. The combined methods employed produce a minimal amount of computational effort as compared to fully resolved grids at the same accuracy. We demonstrate the solver on benchmark problems for the all flow regimes as follows: (1) 2-D backward-facing step using h-adaption, (2) 2-D driven cavity, (3) 2-D natural convection in a differentially heat cavity with h-adaptation, (4) NACA 0012 airfoil in 2-D, (5) supersonic flows over compression ramps, (6) 2-D natural convection in a differentially heat cavity with hp-adaptation, (7) 3-D natural convection in a differentially heat sphere with hp-adaptation. In addition, we show the new moving parts algorithm for working for a 2-D piston; the immersed moving parts method also for valves and pistons, vanes, etc... The movement is performed using an overset grid method and is 2nd order accurate in space, and never produces a tangle grid, that is, robust system at any resolution and any parts configuration. We also show CHT for the currently distributed KIVA-4mpi software and some fairly automatic grid generation using Sandia's Cubit unstructured grid generator. A new electronic web-based manual for KIVA-4 has been developed as well.

02 PETROLEUM↗

Fast algorithms for combustion kinetics calculations: A comparison

To identify the fastest algorithm currently available for the numerical integration of chemical kinetic rate equations, several algorithms were examined. Findings to date are summarized. The algorithms examined include two general-purpose codes EPISODE and LSODE and three special-purpose (for chemical kinetic calculations) codes CHEMEQ, CRK1D, and GCKP84. In addition, an explicit Runge-Kutta-Merson differential equation solver (IMSL Routine DASCRU) is used to illustrate the problems associated with integrating chemical kinetic rate equations by a classical method. Algorithms were applied to two test problems drawn from combustion kinetics. These problems included all three combustion regimes: induction, heat release and equilibration. Variations of the temperature and species mole fraction are given with time for test problems 1 and 2, respectively. Both test problems were integrated over a time interval of 1 ms in order to obtain near-equilibration of all species and temperature. Of the codes examined in this study, only CREK1D and GCDP84 were written explicitly for integrating exothermic, non-isothermal combustion rate equations. These therefore have built-in procedures for calculating the temperature.

Radhakrishnan, K.↗

FDEM v.1.0

Algorithm FOEM numerically solves a system of six, coupled, first-order, linear, inhomogeneous partial differential equations (called the "EH-system") with an explicit, time-domain, finite-difference (FD) The six dependent variables three electric field vector components and three magnetic field FOEM vector components) are stored on staggered spatial and temporal grids. Discrete FO operators approximating the partial derivatives possess 2nd-order accuracy in time and Nth-order accuracy in space. The 3D spatial grid is rectangular, and grid intervals are uniform in each coordinate direction. FD formulae for updating the wavefield variables are appropriately non-dimensionalized and normalized so that single-precision arithmetic calculations yield adequate accuracy. This software will be used to generate electromagnetic responses for typical geological/geophysical models of interest in the petroleum, mining, geotechnical, geothermal, hydrological, environmental assessment, and underground storage industries. Additionally, US government agencies (including the military) involved in monitoring of explosions (chemical or nuclear) or remote surveillance of underground facilities may be interested in using algorithm FOEM. Finally, this software may be of use to organizations involved in fundamental research and applications development in the area of electromagnetic wave propagation (e.g. national laboratories, academic departments. industry).

Source record↗

A Positive-Definite, WENO-Limited, High-Order Finite Volume Solver for 2-D Transport on the Cubed Sphere Using an ADER Time Discretization

Modern computer architectures reward added computation if it reduces algorithmic dependence, reduces data movement, increases accuracy/robustness, and improves memory accesses. The driving motive for this study is to develop a numerical algorithm that respects these constraints while improving accuracy and robustness. This study introduces the ADER-DT (Arbitrary DERivatives in time and space-differential transform) time discretization to positive-definite, weighted essentially nonoscillatory (WENO)-limited, finite volume transport on the cubed sphere in lieu of semidiscrete integrators. The cost of the ADER-DT algorithm is significantly improved from previous implementations without affecting accuracy. A new function-based WENO implementation is also detailed for use with the ADER-DT time discretization. While ADER-DT costs about 1.5 times more than a fourth-order, five-stage strong stability preserving Runge-Kutta (SSPRK4) method, it is far more computationally dense (which is advantageous on accelerators such as graphics processing units), and it has a larger effective maximum stable time step. ADER-DT errors converge more quickly with grid refinement than SSPRK4, giving 6.5 times less error in the L ∞ norm than SSPRK4 at the highest refinement level for smooth data. For nonsmooth data, ADER-DT resolves C 0 discontinuities more sharply. For a complex flow field, ADER exhibits less phase error than SSPRK4. In conclusion, improving both accuracy and robustness as well as better respecting modern computational efficiency requirements, we believe the method presented herein is competitive for efficiently transporting tracers over the sphere for applications targeting modern computing architectures.

97 MATHEMATICS AND COMPUTING↗

Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained “blind” ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

36 MATERIALS SCIENCE↗

Measurement of inclusive jet and dijet cross-sections in proton-proton collisions at $$ \sqrt{s}=13 $$ TeV with the ATLAS detector

Inclusive jet and dijet cross-sections are measured in proton-proton collisions at a centre-of-mass energy of 13 TeV. The measurement uses a dataset with an integrated luminosity of 3.2 fb -1 recorded in 2015 with the ATLAS detector at the Large Hadron Collider. Jets are identified using the anti- k t algorithm with a radius parameter value of R = 0 . 4. The inclusive jet cross-sections are measured double-differentially as a function of the jet transverse momentum, covering the range from 100 GeV to 3.5 TeV, and the absolute jet rapidity up to | y | = 3. The double-differential dijet production cross-sections are presented as a function of the dijet mass, covering the range from 300 GeV to 9 TeV, and the half absolute rapidity separation between the two leading jets within | y | < 3, y * , up to y * = 3. Next-to-leading-order, and next-to-next-to-leading-order for the inclusive jet measurement, perturbative QCD calculations corrected for non-perturbative and electroweak effects are compared to the measured cross-sections.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

A collaborative detection approach for internal steam leakage of tyre vulcanization workshop with artificial immune algorithm

In manufacturing industry, internal leakage of steam trap usually results in great steam waste. In particular, internal steam leakage in tyre vulcanization workshop has significant influences on its production safety and energy efficiency. In practice, internal steam leakage problem often is ignored and it tends to be difficult to detect this leakage due to lack of comprehensive flow meter or method. This paper presents a collaborative detection method for this problem in tyre vulcanization workshop with artificial immune algorithm. In the method, internal leakage of steam trap is defined as nonself antigens, and steam pressure differentials between steam pipe and steam rooms (or bladders) are extracted as epitopes of antigens. Furthermore, periodic energy efficiency and steam pressure of vulcanizer as the danger signals are simultaneously detected. Energy efficiency is represented by damaged cells which will be identified firstly for locating the leaking steam straps through antibodies. Furthermore, the self-adaptive danger thresholds for energy efficiency are evaluated through a steam consumption model and the Levenberg–Marquardt back propagation algorithm. An immune-based clustering algorithm aiNet is then adopted to generate antibodies (detectors) for detection on the steam pressure of vulcanizer. Finally, a case study is implemented to validate this method, which shows that collaborative detection method allows locating the specific leaking steam trap and is a feasible tool to reduce steam waste and ensure the safety of steam supply.

42 ENGINEERING↗