Integration Routines Research Articles

How do organizations refine their post‐merger integration routines over consecutive acquisitions?

AbstractThe post‐merger integration (PMI) phase of mergers and acquisitions (M&A) can determine M&A success. Little is known about how organizations refine their PMI routines with accumulated M&A experience. This study proposes a process model of PMI‐routine development, where the outcomes of routines enacted in earlier acquisitions shape routines in subsequent acquisitions. We test the model using 312 instances of routine implementation, across five consecutive acquisitions by the same acquiring company, and make two key scholarly contributions. First, we advance prior research by identifying how a routine's level of complexity moderates the association between routine adherence and routine outcomes. Second, we elucidate the effect of learning from failure—uncovering a learning process with performance feedback as input. Specifically, negative routine outcomes at the focal acquisition strengthened the positive effect of the improvisation (low adherence) enacted in the focal acquisition on the subsequent modifications to the codified routine, which are likely to improve its performance in future acquisitions.

KeldyshQFT: A C++ codebase for real-frequency multiloop functional renormalization group and parquet computations of the single-impurity Anderson model.

We provide a detailed exposition of our computational framework designed for the accurate calculation of real-frequency dynamical correlation functions of the single-impurity Anderson model in the regime of weak to intermediate coupling. Using quantum field theory within the Keldysh formalism to directly access the self-energy and dynamical susceptibilities in real frequencies, as detailed in our recent publication [Ge et al., Phys. Rev. B 109, 115128 (2024)], the primary computational challenge is the full three-dimensional real-frequency dependence of the four-point vertex. Our codebase provides a fully MPI+OpenMP parallelized implementation of the functional renormalization group (fRG) and the self-consistent parquet equations within the parquet approximation. It leverages vectorization to handle the additional complexity imposed by the Keldysh formalism, using optimized data structures and highly performant integration routines. Going beyond the results shown in the previous publication, the code includes functionality to perform fRG calculations in the multiloop framework, up to arbitrary loop order, including self-consistent self-energy iterations. Moreover, implementations of various regulators, such as hybridization, interaction, frequency, and temperature, are supplied.

PyDFT-QMMM: A modular, extensible software framework for DFT-based QM/MM molecular dynamics.

PyDFT-QMMM is a Python-based package for performing hybrid quantum mechanics/molecular mechanics (QM/MM) simulations at the density functional level of theory. The program is designed to treat short-range and long-range interactions through user-specified combinations of electrostatic and mechanical embedding procedures within periodic simulation domains, providing necessary interfaces to external quantum chemistry and molecular dynamics software. To enable direct embedding of long-range electrostatics in periodic systems, we have derived and implemented force terms for our previously described QM/MM/PME approach [Pederson and McDaniel, J. Chem. Phys. 156, 174105 (2022)]. Communication with external software packages Psi4 and OpenMM is facilitated through Python application programming interfaces (APIs). The core library contains basic utilities for running QM/MM molecular dynamics simulations, and plug-in entry-points are provided for users to implement custom energy/force calculation and integration routines, within an extensible architecture. The user interacts with PyDFT-QMMM primarily through its Python API, allowing for complex workflow development with Python scripting, for example, interfacing with PLUMED for free energy simulations. We provide benchmarks of forces and energy conservation for the QM/MM/PME and alternative QM/MM electrostatic embedding approaches. We further demonstrate a simple example use case for water solute in a water solvent system, for which radial distribution functions are computed from 100ps QM/MM simulations; in this example, we highlight how the solvation structure is sensitive to different basis-set choices due to under- or over-polarization of the QM water molecule's electron density.

Fast immersed boundary method based on weighted quadrature

Combining sum factorization, weighted quadrature, and row-based assembly enables efficient higher-order computations for tensor product splines. We aim to transfer these concepts to immersed boundary methods, which perform simulations on a regular background mesh cut by a boundary representation that defines the domain of interest. Therefore, we present a novel concept to divide the support of cut basis functions to obtain regular parts suited for sum factorization. These regions require special discontinuous weighted quadrature rules, while Gauss-like quadrature rules integrate the remaining support. Two linear elasticity benchmark problems confirm the derived estimate for the computational costs of the different integration routines and their combination. Although the presence of cut elements reduces the speed-up, its contribution to the overall computation time declines with h-refinement.

CNOK: A C++ Glauber model code for single-nucleon knockout reactions

A C++ program package was developed to calculate parallel momentum distributions (including the stripping mechanism and the diffractive dissociation mechanism) of the heavy residue (core) in single-nucleon knockout reactions induced by intermediate-energy (around and more than tens of MeV per nucleon) beams of stable and radioactive atomic nuclei. The program implements the Glauber reaction model that is based on the eikonal approximation and the sudden approximation when dealing with the scattering process, and uses a t-ρ-ρ method to build the projectile-target scattering optical potential, where the nucleonic densities of the core and the target are needed as input. The radial wavefunction of the valence nucleon is solved by a built-in bound-wave solver driven by a globally convergent search engine for optimized potential parameters of the valence nucleon to reproduce user-specified separation energies and root-mean-square (rms) radii of the valence nucleon. The YAML file format is adopted to facilitate user input. The program is expected to especially serve the interpretation of data analysis results of single-nucleon knockout from radioactive ion beams (RIBs), where the experimental practitioners may be more conversant with C++. Program summaryProgram Title: CNOKCPC Library link to program files:https://doi.org/10.17632/dmffpbjhsh.1Developer's repository link:https://gitee.com/asiarabbit/cnokLicensing provisions: GPLv3Programming language: C++Nature of problem: This program calculates the single-particle reaction cross sections of single-nucleon removal from stable and radioactive nuclei impinging on a light composite (i.e. non-hydrogen, e.g. carbon and beryllium) target at intermediate energies (around and more than tens of MeV per nucleon), and the parallel momentum (stripping+diffractive dissociation) distributions of the core, for a pure valence configuration (bound core state+valence nucleon single-particle orbit).Solution method: The bound radial wavefunction of the valence nucleon in a Woods-Saxon (WS) potential is solved following the method of shooting to a fitting point, where the wavefunction is integrated with Runge-Kutta method from 0 and some large distance r∞ to a matching point rm near the nuclear surface. The process is iterated to optimize the eigenvalue Enlj by joining the two waves at rm “smoothly”. Most integrals are evaluated with Simpson's rules, while some with Romberg integration. Searching in the (V0,r0) space of the central WS potential to reproduce the user-specified separation energy and rms radius of the valence nucleon is done via a globally convergent Newton-Raphson root-finding routine for nonlinear systems of equations.Additional comments including restrictions and unusual features: User input is read in with YAML parser (yaml-cpp). Global parameters are readily accessed and uniformly modified in a singleton class. General algorithms (integration, interpolation, minimization, and root-finding routines) are implemented with C++ template class or functors, which adapt to different data types and/or function types. Various batch modes are wrapped in the program package to save user time. For instance, in one run, with the batch mode, one can calculate inclusive single-nucleon removal cross section involving many valence configurations, the relevant inclusive core momentum distribution, and similarly for a series of projectile+target combinations in study of systematics, as is frequently encountered in the study of quenching of single-particle-orbit occupancies (spectroscopic factors) in single-nucleon knockout reactions. Unit tests with Catch are widely deployed to facilitate debugging. C++11 support is required for compilers.

Configuration Space Integration for Adsorbate Partition Functions: The Effect of Anharmonicity on the Thermophysical Properties of CO–Pt(111) and CH3OH–Cu(111)

A method for computing anharmonic thermophysical properties for adsorbates on metal surfaces has been extended to include libration, or frustrated rotation. Classical phase space integration is used with Monte Carlo sampling of the configuration space to obtain the partition function of CO on Pt(111) and CH3OH on Cu(111). A minima-preserving neural network potential energy surrogate is used within the integration routines. Direct state counting using discrete variable representation is used to benchmark the results. We find that the phase space integration approach is in excellent agreement with the direct state counting results. Comparison with standard models such as the harmonic oscillator indicates that anharmonicity contributes significantly to the thermodynamic properties of CH3OH on Cu(111). We find that there is also a considerable difference between the harmonic oscillator and phase space integration for CO on Pt(111), although the discrepancy can largely be attributed to the presence of multiple binding sites within the unit cell. We demonstrate that a multisite harmonic oscillator model might be sufficient for CO–Pt(111). A more thorough description of the potential energy surface, which can be achieved with phase space integration, is necessary for weakly bound adsorbates such as CH3OH. The thermophysical properties were used to calculate free energies of adsorption on the respective metals, and subsequently the equilibrium constants and Langmuir isotherms in relevant temperature ranges. The results show that the choice of model to obtain partition functions greatly affects the resulting surface coverages in kinetic models.

Computationally Efficient Simulations of Stochastically Perturbed Nonlinear Dynamical Systems

Abstract A probabilistic approach is needed to address systems with uncertainties arising in natural processes and engineering applications. For computational convenience, however, the stochastic effects are often ignored. Thus, numerical integration routines for stochastic dynamical systems are rudimentary compared to those for the deterministic case. In this work, the authors present a method to carry out stochastic simulations by using methods developed for the deterministic case. Thereby, the well-developed numerical integration routines developed for deterministic systems become available for studies of stochastic systems. The convergence of the developed method is shown and the method's performance is demonstrated through illustrative examples.

Carlomat_4.0, a new version of the general purpose Monte Carlo program

A new version of the general purpose Monte Carlo program carlomat is presented that substantially improves efficiency of the phase space integration by automatic inclusion of parameterizations which map away the t-channel poles and peaks due to soft and collinear photon or gluon emission. The quadruple precision versions of the routines for computation of the helicity amplitudes and phase-space parameterizations have been written and calls to them implemented in the code generation part of the program. This allows to better control numerical stability of the Monte Carlo programs generated, in particular for reactions with a virtual photon or gluon exchange in the t-channel. A new option of generating s-channel kinematics has been added which takes into account peaks due to Feynman propagators of intermediate bosons which decay into on shell final state particles, or of the top quark. It speeds up both the compilation and execution time of the Monte Carlo programs with respect to the kinematics based on topologies of the Feynman diagrams used in versions 2 and 3 of carlomat. To further speed up the execution time, the main routine of the Monte Carlo program has been supplemented with the Message Passing Interface which allows to run the program parallelly on several processors. Program summaryProgram Title:carlomat, version 4.0CPC Library link to program files:https://doi.org/10.17632/9rx84cpvwc.1Developer's repository link:https://www.kk.us.edu.pl/carlomat.htmlLicensing provisions: GPLv3Programming language: Fortran 90/95Supplementary material: ‘instructions.pdf’Journal reference of previous version: Comput. Phys. Commun. 196 (2015) 563Does the new version supersede the previous version?: YesReasons for the new version: Efficient integration over a multi particle phase space can be a challenging task. Phase space parameterizations allowing to smoothen t-channel poles, or peaks due to soft and collinear photon or gluon emission, which were not taken into account in former versions of carlomat, are now included in the multichannel Monte Carlo integration routine automatically generated by the program. This allows to obtain more reliable predictions for quite a number of reactions that are of interest, either as signal or potential sources of background, both in high or low energy particle colliders.Summary of revisions: Phase space integration routines, originally developed in Phys. Rev. D43 (1991) 3619 and Comput. Phys. Commun. 159 (2004) 106, have been brought to the form suitable for implementation in the general purpose Monte Carlo (MC) program. The code generation part of carlomat has been appropriately modified in order to include new phase space parameterizations in the automatically generated multichannel MC integration routine. Quadruple precision versions of the routines for computation of the helicity amplitudes and phase-space parameterizations have been written and calls to them implemented in the code generation part of carlomat. A number of new options have been introduced in the program which allow to control generation of the multichannel integration routine and the process of MC integration. Among them a new option of generating s-channel kinematics has been added which takes into account peaks due to Feynman propagators of intermediate bosons which decay into on shell final state particles, or of the top quark. It speeds up both the compilation and execution time of the Monte Carlo programs with respect to the kinematics based on topologies of the Feynman diagrams used in versions 2 and 3 of the program. To further speed up the execution time, the main routine of the Monte Carlo program has been supplemented with the Message Passing Interface in order to allow for a parallel run of the program on several processors.Nature of problem: Automatic generation of the MC codes for a wide class of reactions that can be measured in current and future accelerators.Solution method: The Fortran 90/95 MC programs are generated with a meta program that is also written in Fortran 90/95. All amenities of the former versions of carlomat are kept in the current version, too.

Managing Digital Integration Routines in Engineering Firms: Cases of Disruptive BIM Cloud Collaboration Protocols

The challenge for engineering firms to align with an unprecedented variation of digital integration agendas across projects is a recent phenomenon. The literature has offered only a partial understanding of this challenge due to the episodic focus on the digital delivery of individual projects or the initial adoption of digital practice in firms. Through a longitudinal ethnographic study of an engineering firm, this paper explored how practitioners act upon the surge of custom digital integration agendas across multiple projects. By looking at and beyond a large portfolio of projects that demanded building information modeling (BIM) cloud collaboration, this study found that disparate digital agendas across projects disrupt digital integration routines, and they have knock-on effects that can overreach project boundaries and impact projects portfolios and firm operations. With this evidence, this research spotlights the shifting loci of information integration in practice, wherein custom agendas pressure practitioners to shift the focus of integration from their permanent practice at firms to temporary workflows in projects. By revealing organizational responses to this phenomenon, this paper introduces the concept of genericity–complexity trade-off in digital integration routines to explain how firms generate routines that address integration either with ad hoc responses in projects or with systemized integrative efforts across all constellations of practice.

Numerical Integration of Two-Dimensional Complex-Valued Functions for the needs of BIEM

Numerical integration is a key problem with numerous applications, including the boundary integral equation method (BIEM). GSL [7] provides powerful implementations of a broad variety of well-known numerical methods and algorithms. However, its routines have certain limitations: the numerical integration based on QUADPACK library [2] is implemented only in the case of one-dimensional functions. In this paper we present an extension of the GSL numerical integration routines for the special case of two-dimensional complex-valued functions. The presented approach is part of our effort to build a BIEM software system for solving a 3D elastodynamics problem for wave propagation in a continuously inhomogeneous half-space.

NESSi: The Non-Equilibrium Systems Simulation package

The nonequilibrium dynamics of correlated many-particle systems is of interest in connection with pump–probe experiments on molecular systems and solids, as well as theoretical investigations of transport properties and relaxation processes. Nonequilibrium Green’s functions are a powerful tool to study interaction effects in quantum many-particle systems out of equilibrium, and to extract physically relevant information for the interpretation of experiments. We present the open-source software package NESSi (The Non-Equilibrium Systems Simulation package) which allows to perform many-body dynamics simulations based on Green’s functions on the L-shaped Kadanoff–Baym contour. NESSi contains the library libcntr which implements tools for basic operations on these nonequilibrium Green’s functions, for constructing Feynman diagrams, and for the solution of integral and integro-differential equations involving contour Green’s functions. The library employs a discretization of the Kadanoff–Baym contour into time N points and a high-order implementation of integration routines. The total integrated error scales up to O(N−7), which is important since the numerical effort increases at least cubically with the simulation time. A distributed-memory parallelization over reciprocal space allows large-scale simulations of lattice systems. We provide a collection of example programs ranging from dynamics in simple two-level systems to problems relevant in contemporary condensed matter physics, including Hubbard clusters and Hubbard or Holstein lattice models. The libcntr library is the basis of a follow-up software package for nonequilibrium dynamical mean-field theory calculations based on strong-coupling perturbative impurity solvers. Program summaryProgram Title: NESSiCPC Library link to program files:http://dx.doi.org/10.17632/973crf9hgd.1Licensing provisions: MPL v2.0Programming language: C++, pythonExternal routines/libraries: cmake, eigen3, hdf5 (optional), mpi (optional), omp (optional)Nature of problem: Solves equations of motion of time-dependent Green’s functions on the Kadanoff–Baym contour.Solution method: Higher-order solution methods of integral and integro-differential equations on the Kadanoff–Baym contour.

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline- (WEB-spline-) based discrete solution for the 2D Navier-Lamé equation of linear elasticity with a different type of mixed boundary condition called CA,B boundary condition. Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

Stepwise Design Methodology and Heterogeneous Integration Routine of Air-Cooled SiC Inverter for Electric Vehicle

Carrying on SiC devices, the air-cooled inverter of the electric vehicle (EV) can eliminate the traditional complicated liquid-cooling system in order to obtain a light and compact performance of the powertrain, which is considered as the trend of next-generation EV. However, the air-cooled SiC inverter lacks strategic design methodology and heterogeneous integration routine for critical components. In this article, a stepwise design methodology is proposed for the air-cooled SiC inverter in the power module, dc-link capacitor, and heat sink levels. In the power module level, an electrical–thermal–mechanical multiphysics model is proposed. The multidimension stress distribution principles in a six-in-one SiC power module are demonstrated. An improved power module is presented and confirmed by using the observed multiphysics design principles. In the dc-link capacitor level, ripple modeling of the inverter and capacitor are created. Considering the tradeoffs among ripple voltage, ripple current, and cost, optimal strategies to determine the material and minimize the capacitance of the dc-link capacitor are proposed. In the heat sink level, thermal resistance of air-cooled heat sink is modeled. Structure and material properties of the heat sink are optimally designed by using a comprehensive electro-thermal analysis. Based on the optimal design results, the prototypes of the customized SiC power module and heterogeneously integrated air-cooled inverter are fabricated. Experimental results are presented to demonstrate the feasibility of the designed and manufactured air-cooled SiC inverter.

Galileo Ionospheric Correction Algorithm: An Optimization Study of NeQuick‐G

AbstractAt present most low‐cost GNSS receivers operate one frequency in the L band. For them one of the largest error contributions is the delay of radio signals in the Ionosphere. NeQuick‐G is the official ionospheric correction algorithm (ICA), which has been adopted for Galileo, the European GNSS Programme. The NeQuick‐G implementation is complex when compared with other ICAs. It is also demanding in terms of computational resources. The Joint Research Centre completed a reference implementation of NeQuick‐G based on the official document “Ionospheric Correction Algorithm for Galileo Single Frequency Users” provided by the European Global Navigation Satellite Systems Agency. The rationale behind the JRC implementation of NeQuick‐G was the intent to write an independent source code from scratch, without using the pseudo‐codes from the reference document and solely relying on the physics descriptions. Using such implementation as baseline, this paper describes an optimization attempt of the official pseudo‐code from an algorithmic perspective. The objective was to reduce the computational load while not sacrificing the performance. The new proposed integration method is able to speed up calculations to 21% and 49% with respect the two official integration algorithms. The overall computational burden depends on the number of operations, which is eventually closely correlated to the number of calls of the ionospheric model. This underlines the quest to find an integration method reducing this number of calls. Moreover, based on the findings of this study, the authors strongly recommend revisiting the convergence control of the integration routines introduced in (European Commission, 2016, https://www.gsc‐europa.eu/sites/default/files/sites/all/files/Galileo_Ionospheric_Model.pdf).

Fast Computational Convolution Methods for Extended Source Effects in Microlensing Light Curves

Abstract Extended source effects can be seen in gravitational lensing events when sources cross critical lines. Such events probe the stellar intensity profile and could be used to measure limb-darkening coefficients to test stellar model predictions. A database of accurately measured stellar profiles is needed to correctly subtract the stellar flux in planetary transient events. The amount of data that is being produced and that will be produced in current and future microlensing surveys, from both space and ground, requires algorithms that can quickly compute light curves for different source-lens configurations. Based on the convolution method, we describe a general formalism to compute those curves for single lenses. We develop approximations in terms of quadratures of elliptic integrals that we integrate by solving the associated first-order differential equations. We construct analytic solutions for a limb darkening and, for the first time, for a parabolic profile that are accurate at the ∼1%–3% and 0.5% level, respectively. These solutions can be computed orders of magnitude faster than other integration routines. They can be implemented in pipelines processing large data sets to extract stellar parameters in real time.

A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains

A time-domain formulation and its numerical implementation of three-dimensional elastic waves in unbounded domains are discussed. To simulate elastic waves in unbounded domains, it is necessary to truncate the originally infinite extent and remove artificial reflections at truncated boundaries. In this work, perfectly matched layer (PML) is used to surround the truncated regular domain and to absorb outgoing waves beyond the truncated boundaries. To derive governing equations for elastic waves in the PML-truncated domain, complex coordinate stretching is applied to equilibrium, constitutive, and compatibility equations in the frequency domain. Then the stretched equations were converted to the time domain by applying the inverse Fourier transform to yield a system of mixed displacement-stress equations. Introducing a mixed finite element method leads to mixed semi-discrete equations of motion which can be integrated in time for displacements and stresses. Numerical experiments show that reflections from truncated boundaries are significantly reduced by using the developed PML method. For the reduction of computational cost in higher dimensions, an MPI-based parallel computation has been performed on assembly and time integration routines of the finite element wave simulation, which shows good scalability.

Analytic computation of the secular effects of encounters on a binary: features arising from second-order perturbation theory

AbstractBinary–single interactions play a crucial role in the evolution of dense stellar systems such as globular clusters. In addition, they are believed to drive black hole (BH) binary mergers in these systems. A subset of binary–single interactions are secular encounters, for which the third body approaches the binary on a relatively wide orbit, and such that it is justified to average the equations of motion over the binary’s orbital phase. Previous works used first-order (FO) perturbation theory to compute the effects of such secular encounters on the binary. However, this approach can break down for highly eccentric binaries, which are important for BH binary mergers and gravitational wave sources. Here, we present an analytic computation using second-order perturbation techniques, valid to the quadrupole-order approximation. In our calculation, we take into account the instantaneous back reaction of the binary to the third body, and compute corrections to previous FO results. Using singly averaged and direct three-body integrations, we demonstrate the validity of our expressions. In particular, we show that the eccentricity change for highly eccentric binaries can reach a plateau, associated with a large inclination change, and can even reverse sign. These effects are not captured by previous FO results. We provide a simple script to conveniently evaluate our analytic expressions, including routines for numerical integration and verification.

Integration of Palliative Care and Infection Management at the End of Life in U.S. Nursing Homes

ContextInfections in nursing home (NH) residents are often terminal illnesses. Integration of palliative care (PC) and infection management (IM) is a new concept that can help reduce burdensome treatments and improve quality of care for NH residents at the end of life. ObjectivesTo develop measures of integration, describe the integration in U.S. NHs, and examine predictors of integration. MethodsA nationally representative sample of NHs was surveyed. An instrument to measure integration was tested using factor analyses. Descriptive analyses of each integration factor were conducted, construct validity was examined using correlations between the integration factors and validated measures of PC and IM, and multivariable linear regression models were developed to identify NH characteristics associated with integration. ResultsA total of 892 NH surveys were returned (49% response rate), 859 with complete data. Three integration factors were identified: patient involvement in care planning (Involvement), formalized advance care planning (Advance Care Planning), and routine practices of integration (Routine Practices). The highest level of integration in NHs was reported for Involvement (mean (μ) = 73.2, standard error [SE] = 1.57), with lower rates for Advance Care Planning and Routine Practices (respectively, μ = 34.1, SE = 1.05; μ = 31.4, SE = 1.48). Each integration measure was weakly, positively associated with the PC and IM measures (r ≤ 0.25, P ≤ 0.01). There were few associations between NH characteristics and integration. ConclusionIntegration is a distinct concept that is associated with, but different from, PC and IM. Results serve as a baseline assessment of integration in NHs. Continued refinement of the integration instrument is recommended, as is studying if higher integration leads to better resident outcomes.

Vibration Suppression and Defect Detection Schemes in 1D Linear Spring-Mass Systems

In this paper, we present strategies for active vibration suppression and defect detection in a one-dimensional network of an arbitrary number of coupled spring–mass units connected in series. The choice of a spring–mass system is not arbitrary, as the latter is found in many applications throughout a wide range of fields, for instance in defense detection/shielding studies, biomedical engineering, structures engineering, computer graphics and acoustics among others. The system of differential equations that model the spring–mass systems was analyzed and solved using the Laplace transform and other analytic tools. The data used in the numerical simulations were obtained by solving the associated forward problems analytically or numerically. Some of the simulations required numerical integration and minimization routines. A scheme for active vibration suppression is given via explicit formulas for the required control forces. The detect defection strategy is given in terms of an explicit formula whenever only the location or mass of a lone defect is unknown and in terms of a minimization procedure whenever more than one information about the defect(s) are unknown. Several numerical simulations were done to validate these results. As we show in the paper, the success of the vibration suppression scheme we developed depends on the speed and accuracy of the intervening active controls. Meanwhile, the defect detection algorithm only requires measurements in a sufficiently large time interval of the longitudinal vibrations in the first mass.

Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces

Abstract Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of H 1 {H^{1}} , H ⁢ ( curl ) {H(\operatorname{curl})} , H ⁢ ( div ) {H(\operatorname{div})} , and L 2 {L^{2}} inner products, have the 𝒪 ⁢ ( p 7 ) {\mathcal{O}(p^{7})} computational complexity in contrast to the 𝒪 ⁢ ( p 9 ) {\mathcal{O}(p^{9})} cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of H 1 {H^{1}} , H ⁢ ( div ) {H(\operatorname{div})} and H ⁢ ( curl ) {H(\operatorname{curl})} test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.