Subdomains Of Interest Research Articles

A parallel finite element post-processing algorithm for the damped Stokes equations

This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method and post-processing technique. The most valuable highlights of the present algorithm are that (1) a global continuous approximate solution is generated via the partition of unity method; (2) by adding an extra coarse grid correction step, the smoothness of the approximate solution is improved; (3) it has a good parallel performance since there requires little communication in solving a series of residual problems in the subdomain of interest. We theoretically derive the L2-error estimates both for the approximate velocity and pressure and H1-error estimate for the velocity under some necessary conditions. Meanwhile, we numerically perform various test examples to validate the theoretically predicted convergence rate and illustrate the high efficiency of the proposed algorithm.

Bayesian monitoring of substructures under unknown interface assumption

Structural Health Monitoring (SHM) enables assessing in-service structures’ performance by localizing structural anomaly instances immediately after their occurrence. Typical SHM approaches monitor the entire structural spatial domain aggravating the required density and cost of instrumentation. Further, with model-based approaches, the entire structural domain is needed to be defined with high dimensional, compute-intensive models rendering the SHM approaches ill-posed and slow especially when the instrumentation is limited and system observability is compromised. Moreover, in absence of high-fidelity models, oversimplification and subsequent model inaccuracies may lead to inaccurate estimation and possibly false alarms even if a subdomain is modeled inaccurately, e.g. support boundaries. To mitigate such issues, stand-alone monitoring focusing only on a subdomain of interest may be a computationally cheaper and prompt approach while being substantially robust to false alarms. Typically, such stand-alone substructure monitoring approaches demand extensive measurement of the interface, which can be a challenge in real-life applications. This paper presents a novel filtering-based online time domain approach for estimating substructure parameters without the need to measure or estimate the substructure interfaces. The proposed component-wise estimation is stand-alone so that the health estimation of the complete structural domain can be undertaken in parallel and later coupled through post-processing. The requirement of the interface measurement has been alleviated by employing an output injection approach. The proposal has been validated on a numerical beam structure subjected to arbitrary forces and subsequently, the sensitivity against noise and damage severity of the proposal has been investigated. Finally, the proposal is validated on a real beam to illustrate its real-life applicability and significance.

Guaranteed local error estimation for finite element solutions of boundary value problems

This paper considers the finite element solution of the boundary value problem of Poisson’s equation and proposes a guaranteed local error estimation based on the hypercircle method. Compared to the existing literature on qualitative error estimation, the proposed error estimation provides an explicit and sharp bound for the approximation error in the subdomain of interest, and its efficiency can be enhanced by further utilizing a non-uniform mesh. Such a result is applicable to problems without H2-regularity, since it only utilizes the first order derivative of the solution. The efficiency of the proposed method is demonstrated by numerical experiments for both convex and non-convex 2D domains with uniform or non-uniform meshes.

Site-directed late-stage diversification of macrocyclic nannocystins facilitating anticancer SAR and mode of action studies.

Nannocystins are a family of 21-membered cyclodepsipeptides with excellent anticancer activity. However, their macrocyclic architecture poses a significant challenge to structure modification. Herein, this issue is addressed by leveraging the strategy of post-macrocyclization diversification. In particular, a novel serine-incorporating nannocystin was designed so that its appending hydroxyl group could diversify into a wide variety of side chain analogues. Such effort facilitated not only structure-activity correlation at the subdomain of interest, but also the development of a macrocyclic coumarin-labeled fluorescence probe. Uptake experiments indicated good cell permeability of the probe, and endoplasmic reticulum was identified as its subcellular localization site.

A mesh refinement scheme for fourth order bi-harmonic equation of mixed boundary-value elastic problems

Fourth order bi-harmonic equation is extensively used for stress-strain analysis of mixed boundary-value elastic problems. However, currently existing uniform mesh scheme based on finite difference method (FDM) needs vast amount of computational resources and efforts for an acceptable solution. Therefore, in this study, a mesh refinement (MR) scheme based on FDM is developed to solve fourth order bi-harmonic equation effectively. The developed MR scheme allows high resolution computation in sub-domains of interest and relatively low resolution in other regions which overcomes the memory exhausting problems associating with the traditional uniform mesh based FDM. In this paper, sub-domain that needs high resolution (mesh refinement) are identified based on gradient of stress and displacement vectors. A very high gradient in any region signifies the need of fine mesh because coarse grained meshes are not adequate to capture the sharply changing stresses or displacements. Once the sub-domains of interest are identified, the mesh refinement is done by splitting course meshes into smaller meshes. Several new stencils are created to satisfy the fourth order by harmonic equation and associated boundary conditions over the various sizes of meshes. The developed MR scheme has been applied to solve several classical mixed boundary-value elastic problems to show its applicability. In addition, the validity, effectiveness, and superiority of the MR scheme have been established by comparing of obtained solutions with uniform mesh results, finite element method (FEM) results, and the well-known analytical results. Our results show that the developed MR scheme can provide a more reliable and accurate result than the conventional uniform mesh scheme with a reduced number of equations, thus, saves a huge amount of computational memory.

UWB Beam-Based Local Diffraction Tomography—Part II: The Inverse Problem

This two-part article is concerned with the medium reconstruction using the beam-based tomographic inverse scattering. This Part II is based on the results of Part I that has dealt with the preprocessing phase. Specifically, we defined there the beam-frame representation of the scattered field and the corresponding processing windows that transform the scattering data to the beam domain. We have also derived the “local time-domain diffraction tomography” relation according to which the beam-domain data are directly related to the local Radon transform (LRT) of the medium. This local transform can be inverted and is used here for the local reconstruction of the medium via beam-domain filtered backpropagation. In this article, we define the filtered backpropagated beam waves and then reconstruct the medium in any sub-domain of interest (DoI) by aggregating the contributions of the backpropagated beams that pass in or near that DoI. Specifically, we use the class of isodiffracting beam waves, namely, the isodiffracting Gaussian beams (ID-GBs) and the isodiffracting pulse beams (ID-PBs) for the frequency- and time-domain formulations, respectively. Explicit expressions for the filtered backpropagated reconstruction kernels are given. The efficacy of the beam-domain approach for local backpropagation and reconstruction is demonstrated via numerical examples of the synthetic noisy data.

A parallel stabilized finite element variational multiscale method based on fully overlapping domain decomposition for the incompressible Navier-Stokes equations

Based on a fully overlapping domain decomposition approach, a parallel stabilized finite element variational multiscale method for the incompressible Navier-Stokes equations is proposed, where the stabilizations both for the velocity and pressure are based on two local Gauss integrations at the element level. The basic idea of the method is to use a locally refined global mesh to compute a stabilized solution in the given subdomain of interest. The proposed method only requires the application of an existing Navier-Stokes sequential solver on the locally refined global mesh associated with each subdomain, and thus can reuse the existing sequential solver without substantial recoding. Error bound of the approximate solutions from the proposed method is estimated with the use of local a priori error estimate for the stabilized solution. Algorithmic parameter scalings of the method are also derived. Some numerical simulations are presented to demonstrate the effectiveness of the method.

Enabling numerically exact local solver for waveform inversion—a low-rank approach

As in many fields, in seismic imaging, the data in the field is collected over a relatively large medium even though only a part of that medium is truly of interest. This results in significant waste in computation as a typical inversion algorithm requires many solutions of the wave equation throughout the entire domain, even if only a small part of the domain is being updated. One way to mitigate this is to use a numerically exact local wave equation solver to perform waveform inversions in an area of interest, where the idea is to compute accurate solutions of the wave equation within a subdomain of interest. Although such solvers exist, many require the computation of Green’s function matrices in the background domain. For large-scale seismic data acquisition, the computation of the Green’s function matrices is prohibitively expensive since it involves solving thousands of partial differential equations in the background model. To mitigate this, in this work, we propose to exploit the low-rank structure of the full subsurface Green’s function. Using carefully selected 2D stylized models, we first show that the full subsurface Green’s function tensor organized as a matrix exhibits the low-rank structure in a transform domain. We then propose a randomized singular value decomposition–based framework to compute the low-rank approximation of the Green’s functions, where the cost of wave equation solves depends on the rank of the underlying Green’s function matrix instead of the number of grid points at the surface of the background model and on the boundary of the local domain. Next, we validate the proposed framework by performing time-lapse waveform inversion using the 2D Marmousi model. Finally, we demonstrate a rank-minimization–based framework to compute the low-rank factorized form of the Green’s function matrices in large-scale 3D seismic data acquisition.

Two-dimensional local Fourier image reconstruction via domain decomposition Fourier continuation method.

The MRI image is obtained in the spatial domain from the given Fourier coefficients in the frequency domain. It is costly to obtain the high resolution image because it requires higher frequency Fourier data while the lower frequency Fourier data is less costly and effective if the image is smooth. However, the Gibbs ringing, if existent, prevails with the lower frequency Fourier data. We propose an efficient and accurate local reconstruction method with the lower frequency Fourier data that yields sharp image profile near the local edge. The proposed method utilizes only the small number of image data in the local area. Thus the method is efficient. Furthermore the method is accurate because it minimizes the global effects on the reconstruction near the weak edges shown in many other global methods for which all the image data is used for the reconstruction. To utilize the Fourier method locally based on the local non-periodic data, the proposed method is based on the Fourier continuation method. This work is an extension of our previous 1D Fourier domain decomposition method to 2D Fourier data. The proposed method first divides the MRI image in the spatial domain into many subdomains and applies the Fourier continuation method for the smooth periodic extension of the subdomain of interest. Then the proposed method reconstructs the local image based on L2 minimization regularized by the L1 norm of edge sparsity to sharpen the image near edges. Our numerical results suggest that the proposed method should be utilized in dimension-by-dimension manner instead of in a global manner for both the quality of the reconstruction and computational efficiency. The numerical results show that the proposed method is effective when the local reconstruction is sought and that the solution is free of Gibbs oscillations.

A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken’s dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.

Millimeter-wave spotlight imager using dynamic holographic metasurface antennas.

Computational imaging systems leverage generalized measurements to produce high-fidelity images, enabling novel and often lower cost hardware platforms at the expense of increased processing. However, obtaining full resolution images across a large field-of-view (FOV) can lead to slow reconstruction times, limiting system performance where faster frame rates are desired. In many imaging scenarios, the highest resolution is needed only in smaller subdomains of interest within a scene, suggesting an aperture supporting multiple modalities of image capture with different resolutions can provide a path to system optimization. We explore this concept in the context of millimeter-wave imaging, presenting the design and simulation of a single frequency (75 GHz), multistatic, holographic spotlight aperture integrated into a K-band (17.5-26.5 GHz), frequency-diverse imager. The spotlight aperture - synthesized using an array of dynamically tuned, holographic, metasurface antennas - illuminates a constrained region-of-interest (ROI) identified from a low-resolution image, extracting a high-fidelity image of the constrained-ROI with a minimum number of measurement modes. The designs of both the static, frequency-diverse sub-aperture and the integrated dynamic spotlight aperture are evaluated using simulation techniques developed for large-scale synthetic apertures.

Reduced order variational multiscale enrichment method for elasto-viscoplastic problems

This manuscript presents the formulation and implementation of a novel reduced order variational multiscale enrichment (ROVME) method for elasto-viscoplastic problems. This method provides a hierarchical model order reduction technique based on the eigenstrain concept to approximate the fine scale response resolved at subdomains of interest. By eliminating the requirement of direct fine scale discretization, the computational effort associated with the variational multiscale enrichment (VME) method is significantly reduced. The model order reduction is achieved in the scale-coupled inelastic problem by automatically satisfying the microscale equilibrium state through the eigenstrain concept and coarse discretization of inelastic strain fields within the microscale domain. The inelastic material behavior is idealized with coupled Perzyna type viscoplasticity and flow stress evolution based on the Johnson–Cook model. Numerical verifications are performed to assess the capabilities of the proposed methodology, against the direct VME method with detailed fine scale resolutions. The verification results demonstrate high accuracy and computational efficiency of the reduced order VME framework for elasto-viscoplastic problems with material heterogeneity.

A–[formula omitted] finite element method with composite grids for time-dependent eddy current problem

We investigate the A–ϕ finite element method with the global coarse grids and the local fine grids (composite grids) for solving a time-dependent eddy current problem, which can improve the accuracy of the coarse grid solutions in some subdomains of interest in the case when properly increasing computational costs. Meanwhile, to decrease calculation complexity and avoid dealing with a saddle-point problem from the traditional A–ϕ scheme, we design an iteration which combines the composite grid method with classic steepest descent such that the solutions of the unknowns A and ϕ for the global coarse grids domain are decoupled in two separate equations. We prove it converges with a bounded rate independent of the mesh sizes.

Adaptive meshing for finite element analysis of heterogeneous materials

Adaptive meshing of 2D planar regions, curved surfaces as well as 3D volumes has been extensively studied in Finite Element Analysis (FEA) in the past few decades. Despite the maturity of these adaptive meshing approaches, most of the existing schemes assume the domain or sub-domain of interest is homogeneous. In the context of FEA of heterogeneous objects, traditional adaptive mesh generation strategies become inadequate as they fail to take into account the material heterogeneities. This paper is motivated to tackle such problems and propose an adaptive mesh generation scheme for FEA of versatile heterogeneous materials. The proposed approach takes full advantages of the material heterogeneity information, and the mesh density is formulated with a specific function of the material variations. Dual triangulation of centroidal Voronoi tessellation is then constructed and necessary mesh subdivision is applied with respect to a predefined material threshold. Experiments show that the proposed approach distributes the material composition variation over mesh elements as equally as possible and thus minimizes the number of elements in terms of the given material threshold. FEA results show that the proposed method can significantly decrease the mesh complexities as well as computational resources in FEA of heterogeneous objects and compared with existing approaches, significant mesh reduction can be achieved without sacrifice in FEA qualities.

Bayesian compressive sensing for ultrawideband inverse scattering in random media

We develop an ultrawideband inverse scattering technique for reconstructing continuous random media on the basis of Bayesian compressive sensing. In addition to providing maximum a posteriori estimates of the unknown weights, Bayesian inversion provides an estimate of the confidence level of the solution, as well as a systematic approach for optimizing subsequent measurement(s) to maximize information gain. We impose sparsity priors directly on spatial harmonics to exploit the spatial correlation exhibited by continuous media and then efficiently solve for their posterior probability density functions by using a fast relevance vector machine. We linearize the problem by using the first-order Born approximation, which enables us to combine, in a single inversion, measurements from multiple transmitters and ultrawideband frequencies. To extend the method to high-contrast media, we also introduce a Bayesian compressive sensing distorted-Born iterative approach. Further, we illustrate the use of time-reversal strategies to adaptively focus the inversion effort onto subdomains of interest and hence reduce the overall inversion cost. The proposed techniques are applied to a number of scenarios including crosshole and borehole sensing.

Logistic analysis of epidemiologic studies with augmentation sampling involving re-stratification and population expansion.

Epidemiologic cross-sectional, case-cohort, or case-control studies often select augmentation samples to supplement an existing (baseline) sample, primarily for the two reasons: (1) to increase the sample sizes from certain subdomains of interest that were not originally considered in the design of the baseline study and (2) to obtain samples from an extension of the target population. To address these two objectives, two-stage stratified sample designs are considered, where the stratification based on the expanded population at the second stage is not nested in the first stage strata. The sample weighting and Taylor linearization variance estimation for the two-stage stratified sample designs, involving re-stratification and population expansion, are provided for estimating population totals and logistic regression coefficients. Results from limited simulation studies and a logistic regression analysis of a study of human papillomavirus serology are provided.

MUSIC Imaging Using Phase-Space Gaussian-Beams Processing

In a recent paper we presented a beam-based backpropagation and correlation imaging scheme for targets in a homogeneous medium. This beam-based approach is extended here for MUSIC-imaging algorithms. In the beam approach, the fields of the physical arrays of point-sources and point receivers are expanded using special sets of collimated beam-sources and beam-receivers. This converts the physical scattering data into a beam-domain data, describing the scattering amplitudes seen (synthetically) by receiver beams due to excitation by source beams. The image is then formed by applying the MUSIC algorithm directly in the beam domain. We derive a closed form expression for the data transformation to the beam domain and then derive multi-experiments MUSIC-imaging algorithms that accommodate the measurement noise. The beam approach enables local imaging of any given sub-domain of interest by retaining only the subset of beams that pass through that domain, thus reducing the overall computation complexity relative to the conventional Green function approach, and filtering out of data and noise that arrive from non-relevant directions. Numerical investigations reveal that the beam approach also provides a better resolution under low S/N conditions.

Back-Propagation and Correlation Imaging Using Phase-Space Gaussian-Beams Processing

We present a beam-based imaging scheme for targets in homogeneous medium whereby the fields of the source and receiver arrays are expanded using special phase-space bases of collimated beam fields, thus converting the physical data into a beam-domain data describing the scattering amplitudes seen (synthetically) by the receiver beams due to excitation by the source beams. The image is then formed by correlating the backpropagated beam data with the incident beams. The formulation utilizes the ultra-wideband phase-space beam-summation method where the beam bases consist of Gaussian beams that emerge from a discrete set of points and directions in the source and the receiver domains. An important feature of this method is that the beam-sets are frequency independent and hence are calculated once and then used for all frequencies. A closed form expression for the data-transformation matrix from the physical domain to the beam domain is derived, leading to sparse beam-domain data. The beam approach enables local imaging of any sub-domain of interest by retaining only the subset of source and receiver beams that pass through that domain, thus reducing the overall computation complexity. The method properties are explored via numerical simulations in a noisy environment.

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

In this paper, we analyse local properties of the continuous interior penalty (CIP) method for a model convection-dominated singularly perturbed convection–diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows thats locally, the CIP method is comparable to the streamline-diffusion or the discontinuous Galerkin methods.

Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quasilinear elliptic problems

Two types of pointwise a posteriori error estimates are presented for gradients of finite element approximations of second-order quasilinear elliptic Dirichlet boundary value problems over convex polyhedral domains $\Omega$ in space dimension n \ge 2. We first give a residual estimator which is equivalent to \|\nabla(u-u_h)\|_{L_\infty(\Omega) up to higher-order terms. The second type of residual estimator is designed to control \nabla(u-u_h)locally over any subdomain of $\Omega$. It is a novel a posteriori counterpart to the localized or weighted a priori estimates of [Sch98]. This estimator is shown to be equivalent (up to higher-order terms) to the error measured in a weighted global norm which depends on the subdomain of interest. All estimates are proved for general shape-regular meshes which may be highly graded and unstructured. The constants in the estimates depend on the unknown solution u in the nonlinear case, but in a fashion which places minimal restrictions on the regularity of u.