Isogeometric collocation method for the fractional Laplacian in the 2D bounded domain
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Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
Here, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy λ ≥ μ to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.
42 ENGINEERING↗
Asymptotically Compatible Reproducing Kernel Collocation and Meshfree Integration for Nonlocal Diffusion
Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The numerical scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for a linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme, which is stable. In addition, assembling the stiffness matrix of the nonlocal problem requires costly computational resources because high-order Gaussian quadrature is necessary to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares.
97 MATHEMATICS AND COMPUTING↗
Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations
Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.
97 MATHEMATICS AND COMPUTING↗
Computational Fluid Dynamics Using the Adaptive Wavelet-Collocation Method
Advancements to the adaptive wavelet-collocation method over the last decade have opened up a number of new possible areas for active research. Volume penalization techniques allow complex immersed boundary conditions to be used with high efficiency for both internal and external flows. Anisotropic methods make it possible to use body-fitted meshes while still taking advantage of the dynamic adaptability properties wavelet-based methods provide. The parallelization of the approach has made it possible to perform large high-resolution simulations of detonation initiation and fluid instabilities to uncover new physical insights that would otherwise be difficult to discover. Other developments include space-time adaptive methods and nonreflecting boundary conditions. This article summarizes the work performed using the adaptive wavelet-collocation method developed by Vasilyev and coworkers over the past decade.
42 ENGINEERING↗
A Hybrid Lagrangian/Eulerian Collocated Velocity Advection and Projection Method for Fluid Simulation
Abstract We present a hybrid particle/grid approach for simulating incompressible fluids on collocated velocity grids. Our approach supports both particle‐based Lagrangian advection in very detailed regions of the flow and efficient Eulerian grid‐based advection in other regions of the flow. A novel Backward Semi‐Lagrangian method is derived to improve accuracy of grid based advection. Our approach utilizes the implicit formula associated with solutions of the inviscid Burgers’ equation. We solve this equation using Newton's method enabled by C 1 continuous grid interpolation. We enforce incompressibility over collocated, rather than staggered grids. Our projection technique is variational and designed for B‐spline interpolation over regular grids where multiquadratic interpolation is used for velocity and multilinear interpolation for pressure. Despite our use of regular grids, we extend the variational technique to allow for cut‐cell definition of irregular flow domains for both Dirichlet and free surface boundary conditions.
Gagniere, S.↗
Analysis of Three Types of Collocated Disdrometer Measurements at the ARM Southern Great Plains Observatory
To better provide U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) user facility’s disdrometer deployment strategy and support ARM precipitation-related projects, this study analyzes 18 months of drop size distribution (DSD) observations from six collocated disdrometers deployed at the ARM Southern Great Plains (SGP) site. Emphasis is placed on quantifying the uncertainties related to DSD and rainfall properties using different types of disdrometers and different pairing concepts. The instrument sampling errors are also discussed, which improves our understanding of instrument-specific impacts on rainfall measurements. The key findings are as follows: 1) All the disdrometers show consistent behaviors overall regarding accumulated precipitation, mean rainfall rate, DSD parameters, and radar reflectivity values; and 2) Strong agreement is shown between three disdrometer types in terms of their DSDs for mid-size drop range (D = 1 mm-4 mm). The two-dimensional video disdrometer performance meets expectations as the most accurate unit among all. The paired disdrometer shows a reduced statistical sampling error in terms of drop counts and other DSD properties. A concept of deploying two or more collocated disdrometers side by side is recommended for future ARM field campaigns.
54 ENVIRONMENTAL SCIENCES↗
Investigating Inherent Numerical Stabilization for the Moist, Compressible, Non‐Hydrostatic Euler Equations on Collocated Grids
This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite-Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth-order-accurate reconstructions with and without Weighted Essentially Non-Oscillatory limiting. Acoustic-only upwinding for pressure and mass flux terms and advective-only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st-order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k −5/3 to smaller spatial scales by reducing dissipation either with higher-order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k −5/3 exhibit physically realizable solutions, even though the spectra appear to be physical.
54 ENVIRONMENTAL SCIENCES↗
Microstructure-Sensitive Uncertainty Quantification for Crystal Plasticity Finite Element Constitutive Models Using Stochastic Collocation Methods
Uncertainty quantification (UQ) plays a major role in verification and validation for computational engineering models and simulations, and establishes trust in the predictive capability of computational models. In the materials science and engineering context, where the process-structure-property-performance linkage is well known to be the only road mapping from manufacturing to engineering performance, numerous integrated computational materials engineering (ICME) models have been developed across a wide spectrum of length-scales and time-scales to relieve the burden of resource-intensive experiments. Within the structure-property linkage, crystal plasticity finite element method (CPFEM) models have been widely used since they are one of a few ICME toolboxes that allows numerical predictions, providing the bridge from microstructure to materials properties and performances. Several constitutive models have been proposed in the last few decades to capture the mechanics and plasticity behavior of materials. While some UQ studies have been performed, the robustness and uncertainty of these constitutive models have not been rigorously established. In this work, we apply a stochastic collocation (SC) method, which is mathematically rigorous and has been widely used in the field of UQ, to quantify the uncertainty of three most commonly used constitutive models in CPFEM, namely phenomenological models (with and without twinning), and dislocation-density-based constitutive models, for three different types of crystal structures, namely face-centered cubic (fcc) copper (Cu), body-centered cubic (bcc) tungsten (W), and hexagonal close packing (hcp) magnesium (Mg). Our numerical results not only quantify the uncertainty of these constitutive models in stress-strain curve, but also analyze the global sensitivity of the underlying constitutive parameters with respect to the initial yield behavior, which may be helpful for robust constitutive model calibration works in the future.
36 MATERIALS SCIENCE↗
High Dynamic Range AFM Cantilever With a Collocated Piezoelectric Actuator-Sensor Pair
We present a novel approach to fundamentally eliminate electrical feedthrough (cross-talk) signal in self-sensing piezoelectric microcantilevers, used in dynamic mode atomic force microscopy (AFM). The probe is an Si microcantilever on which a two-layer piezoelectric stack transducer is microfabricated. The top transducer in the stack functions as an actuator, while the bottom transducer is used as a deflection sensor. This arrangement leads to a collocated actuator-sensor pair that co-exist on the same surface area. We report experimental results that establish the sensor's ability to capture the full dynamic range of the cantilever with minimal feedthrough from the actuator.
42 ENGINEERING↗
Solving Eco-Driving Problems Using Indirect Collocation Method and Smooth Representation
Here, we discuss the eco-driving problem, considering both electric and conventional powertrains, and presents a pathway to solving it numerically using an indirect collocation method. Despite the low-order system dynamics, the piecewise fuel/efficiency map, gear shifting, and real-world traffic/road situations bring system discontinuities/switchings and pure state constraints into the problem formulation, which make the problem highly nonlinear and nontrivial to solve. This letter introduces smooth approximations to convert the original problem to an unconstrained (and penalized) smooth boundary-value problem. This approach eliminates the discussion of the switching structure and leads to a lightweight Newton-method-based solution procedure.
42 ENGINEERING↗
Application of Ocean Thermal Energy Conversion (OTEC) Systems for Powering Safety Monitoring Systems of Offshore Oil and Gas Operations, OESI 2.0 M-1 T-1-P1.1 – Objective 3 Report: Collocation of OTEC with Offshore Oil and Gas Activities
This report is a component of a comprehensive research initiative aimed at evaluating the technoeconomic feasibility of deploying Ocean Thermal Energy Conversion (OTEC) systems to power safety monitoring systems for offshore oil and gas operations in the Gulf of America (GOA). The overarching study has three primary objectives: summarizing existing oil and gas equipment monitoring systems, modeling OTEC as a renewable marine energy source, and exploring the collocation of OTEC power sources with offshore oil and gas activities to offset power demand. This report specifically addresses Objective 3, focusing on the integration of OTEC systems with floating oil and gas facilities.
02 PETROLEUM↗
Identifying a regional aerosol baseline in the eastern North Atlantic using collocated measurements and a mathematical algorithm to mask high-submicron-number-concentration aerosol events
Abstract. High-time-resolution measurements of in situ aerosol and cloud properties provide the ability to study regional atmospheric processes that occur on timescales of minutes to hours. However, one limitation to this approach is that continuous measurements often include periods when the data collected are not representative of the regional aerosol. Even at remote locations, submicron aerosols are pervasive in the ambient atmosphere with many sources. Therefore, periods dominated by local aerosol should be identified before conducting subsequent analyses to understand aerosol regional processes and aerosol–cloud interactions. Here, we present a novel method to validate the identification of regional baseline aerosol data by applying a mathematical algorithm to the data collected at the U.S. Department of Energy's (DOE) Atmospheric Radiation Measurement (ARM) user facility in the eastern North Atlantic (ENA). The ENA central facility (C1) includes an aerosol observing system (AOS) for the measurement of aerosol physical, optical, and chemical properties at time resolutions from seconds to minutes. A second temporary supplementary facility (S1), located ∼0.75 km from C1, was deployed for ∼1 year during the Aerosol and Cloud Experiments (ACE-ENA) campaign in 2017. First, we investigate the local aerosol at both locations. We associate periods of high submicron number concentration (Ntot) in the fine-mode condensation particle counter (CPC) and size distributions from the Ultra-High Sensitivity Aerosol Spectrometer (UHSAS) as a function of wind direction using a meteorology sensor with local sources. Elevated concentrations of Aitken-mode (< 100 nm diameter) particles were observed in correspondence with the wind directions associated with airport operations. At ENA, the Graciosa Airport and its associated activities were found to be the main sources of high-concentration aerosol events at ENA, causing peaks in 1 min Ntot that exceeded 8000 and 10 000 cm−3 at C1, in summer and winter, respectively, and 5000 cm−3 at S1 in summer. Periods with high Ntot not associated with these wind directions were also observed. As a result, the diverse local sources at ENA yielded a poor relationship between Ntot measurements collected at C1 and S1 (R2=0.03 with a slope =0.05±0.001). As a first approach to mask these events, the time periods when the wind direction was associated with the airport operations (west to northwest and southeast to south at C1 and east to south at S1) were applied. The meteorological masks removed 38.9 % of the data at C1 and 43.4 % at S1, and they did not significantly improve the relationship between the two sites (R2=0.18 with a slope =0.06±0.001). Due to the complexity of high-Ntot events observed at ENA, we develop and validate a mathematical ENA Aerosol Mask (ENA-AM) to identify high-Ntot events using 1 min resolution data from the AOS CPC at C1 and S1. After its parameterization and application, ENA-AM generated a high correlation between Ntot in the summer at C1 and S1 (R2=0.87 with a slope =0.84±0.001). We identified the regional baseline at ENA to be 428±228 cm−3 in the summer and 346±223 cm−3 in the winter. Lastly, we compared masked measurements from the AOS with the ARM Aerial Facility (AAF) during flights over C1 in the summer to understand submicron aerosol vertical mixing over C1. The high correlation (R2=0.71 with a slope of 1.04±0.01) observed between C1 and the AAF Ntot collected within an area of 10 km surrounding ENA and at altitudes < 500 m indicated that the submicron aerosol at ENA was well mixed within the first 500 m of the marine boundary layer during the month of July during ACE-ENA. Our novel method for determining a regional aerosol baseline at ENA can be applied to other time periods and at other locations with validation by a secondary site or additional collocated measurements.
54 ENVIRONMENTAL SCIENCES↗
Tensor Network Space-Time Spectral Collocation Method for Solving the Nonlinear Convection Diffusion Equation
Not Available
97 MATHEMATICS AND COMPUTING↗
Surrogate approximation of the Grad–Shafranov free boundary problem via stochastic collocation on sparse grids
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. Furthermore, the use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
Not Available
Fan, Yiming (ORCID:0000000247830699)↗
Collocation methods for nonlinear differential equations on low-rank manifolds
We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.
97 MATHEMATICS AND COMPUTING↗
AFM Microcantilever With a Collocated AlN Sensor-Actuator Pair: Enabling Efficient Q-Control for Dynamic Imaging
This manuscript presents a novel microcantilever with an embedded piezoelectric sensor-actuator pair for dynamic atomic force microscopy (AFM). The transducer pair is constructed from a two-layered AlN stack. Stacking the piezoelectric transducers in this manner leads to a minimal feedthrough from actuation to sense electrode, granting a high dynamic range frequency response for dynamic mode AFM. The cantilever's design allows for dual mode operation at 1 st and 2 nd resonance modes. High resolution tapping mode imaging results are reported, while the cantilever is operated at these modes. A feedback control loop is used to modify quality factor of the 1 st mode of the cantilever, using a positive position feedback (PPF) controller. Here, a faster response time is achieved by reducing the Q-factor, enabling the cantilever to track the topography at a higher scan rate. Results of bimodal AFM imaging are reported, using amplitude changes of the 1 st mode for surface topography, while material properties are encoded in phase changes of the 2 nd mode.
47 OTHER INSTRUMENTATION↗