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We present a method for the numerical integration of homogeneous functions over convex and nonconvex polygons and polyhedra. On applying Stokes’s theorem and using the property of homogeneous functions, we show that it suffices to integrate these functions on the boundary facets of the polytope. For homogeneous polynomials, this approach is used to further reduce the integration to just function evaluations at the vertices of the polytope. This results in an exact cubature rule for a homogeneous polynomial f , where the integration points are only the vertices of the polytope and the function f and its partial derivatives are evaluated at these vertices. Numerical integration of homogeneous functions in polar coordinates and on curved domains are also presented. Along with an efficient algorithm for its implementation, we showcase several illustrative examples in two and three dimensions that demonstrate the accuracy of the proposed method.

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P. The bounds are in terms of a single geometric quantity h*, which denotes the minimum distance between a vertex of P and any hyperplane containing a nonincident face. We prove that the upper bound is sharp for d = 2 and analyze the bounds in the special cases of hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using MATLAB and employ them in a three-dimensional finite element solution of the Poisson equation on a nontrivial polyhedral mesh. As expected from the upper bound derivation, the H1-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.

In this paper, we provide a retrospective examination of the developments and applications of the extended finite element method (X-FEM) in computational fracture mechanics. Our main attention is placed on the modeling of cracks (strong discontinuities) for quasistatic crack growth simulations in isotropic linear elastic continua. We provide a historical perspective on the development of the method, and highlight the most important advances and best practices as they relate to the formulation and numerical implementation of the X-FEM for fracture problems. Existing challenges in the modeling and simulation of dynamic fracture, damage phenomena, and capturing the transition from continuum-to-discontinuum are also discussed.

We introduce a PDE-based node-to-element contact formulation as an alternative to classical, purely geometrical formulations. It is challenging to devise solutions to nonsmooth contact problem with continuous gap using finite element discretizations. We herein achieve this objective by constructing an approximate distance function (ADF) to the boundaries of solid objects, and in doing so, also obtain universal uniqueness of contact detection. Unilateral constraints are implemented using a mixed model combining the screened Poisson equation and a force element, which has the topology of a continuum element containing an additional incident node. An ADF is obtained by solving the screened Poisson equation with constant essential boundary conditions and a variable transformation. The ADF does not explicitly depend on the number of objects and a single solution of the partial differential equation for this field uniquely defines the contact conditions for all incident points in the mesh. Having an ADF field to any obstacle circumvents the multiple target surfaces and avoids the specific data structures present in traditional contact-impact algorithms. We also relax the interpretation of the Lagrange multipliers as contact forces, and the Courant–Beltrami function is used with a mixed formulation producing the required differentiable result. We demonstrate the advantages of the new approach in two- and three-dimensional problems that are solved using Newton iterations. Simultaneous constraints for each incident point are considered.

We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre’s method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.

Endothelial cell (EC)-enriched protein coding genes, such as endothelial nitric oxide synthase (eNOS), define quintessential EC-specific physiologic functions. It is not clear whether long noncoding RNAs (lncRNAs) also define cardiovascular cell type-specific phenotypes, especially in the vascular endothelium. Here, we report the existence of a set of EC-enriched lncRNAs and define a role for spliced-transcript endothelial-enriched lncRNA (STEEL) in angiogenic potential, macrovascular/microvascular identity, and shear stress responsiveness. STEEL is expressed from the terminus of the HOXD locus and is transcribed antisense to HOXD transcription factors. STEEL RNA increases the number and integrity of de novo perfused microvessels in an in vivo model and augments angiogenesis in vitro. The STEEL RNA is polyadenylated, nuclear enriched, and has microvascular predominance. Functionally, STEEL regulates a number of genes in diverse ECs. Of interest, STEEL up-regulates both eNOS and the transcription factor Kruppel-like factor 2 (KLF2), and is subject to feedback inhibition by both eNOS and shear-augmented KLF2. Mechanistically, STEEL up-regulation of eNOS and KLF2 is transcriptionally mediated, in part, via interaction of chromatin-associated STEEL with the poly-ADP ribosylase, PARP1. For instance, STEEL recruits PARP1 to the KLF2 promoter. This work identifies a role for EC-enriched lncRNAs in the phenotypic adaptation of ECs to both body position and hemodynamic forces and establishes a newer role for lncRNAs in the transcriptional regulation of EC identity.

This paper is an overview of recent developments in the construction of finite element interpolants, which areC ^sup 0^-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants--Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.[PUBLICATION ABSTRACT]

HIV-1-infected persons are at higher risk of lower respiratory tract infections than HIV-1-uninfected individuals. This suggests strongly that HIV-infected persons have specific impairment of pulmonary immune responses, but current understanding of how HIV alters pulmonary immunity is incomplete. Alveolar macrophages (AMs), comprising small and large macrophages, are major effectors of innate immunity in the lung. We postulated that HIV-1 impairs pulmonary innate immunity through impairment of AM physiological functions. AMs were obtained by bronchoalveolar lavage from healthy, asymptomatic, antiretroviral therapy-naive HIV-1-infected and HIV-1-uninfected adults. We used novel assays to detect in vivo HIV-infected AMs and to assess AM functions based on the HIV infection status of individual cells. We show that HIV has differential effects on key AM physiological functions, whereby small AMs are infected preferentially by the virus, resulting in selective impairment of phagocytic function. In contrast, HIV has a more generalized effect on AM proteolysis, which does not require direct viral infection. These findings provide new insights into how HIV alters pulmonary innate immunity and the phenotype of AMs that harbors the virus. They underscore the need to clear this HIV reservoir to improve pulmonary immunity and reduce the high incidence of lower respiratory tract infections in HIV-1-infected individuals.

For an integrand with a 1/ r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p ( x )/ r α , where p is a trivariate polynomial and α > 0 is the strength of the singularity. We use the map ( u , v , w ) → ( x , y , z ) : x = u β , y = x v , z = x w , and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation ( β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

We demonstrate applications of quantitative structure–property relationship (QSPR) modeling to supplement first-principles computations in materials design. We have here focused on the design of polymers with specific electronic properties. We first show that common materials properties such as the glass transition temperature ( T g ) can be effectively modeled by QSPR to generate highly predictive models that relate polymer repeat unit structure to T g . Next, QSPR modeling is shown to supplement and guide first-principles density functional theory (DFT) computations in the design of polymers with specific dielectric properties, thereby leveraging the power of first-principles computations by providing high-throughput capability. Our approach consists of multiple rounds of validated MQSPR modeling and DFT computations to optimize the polymer skeleton as well as functional group substitutions thereof. Rigorous model validation protocols insure that the statistical models are able to make valid predictions on molecules outside the training set. Future work with inverse QSPRs has the potential to further reduce the time to optimize materials properties.