When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations.

We consider a Markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes. Customers waiting to be served may abandon the queue, and there is a cost penalty associated with such abandonments. Service rates, abandonment rates and abandonment penalties are generally different for the different classes. The problem studied is that of dynamically scheduling the various classes. We consider the Halfin-Whitt heavy traffic regime, where the total arrival rate and the number of servers both become large in such a way that the system’s traffic intensity parameter approaches one. An approximating diffusion control problem is described and justified as a purely formal (i.e., non rigorous) heavy traffic limit. The Hamilton-Jacobi-Bellman equation associated with the limiting diffusion control problem is shown to have a smooth (classical) solution, and optimal controls are shown to have an extremal or “bang-bang ” character. Several useful qualitative insights are derived from the mathematical analysis, including a “square root rule ” for sizing large systems and a sharp contrast between system behavior in the Halfin-Whitt regime versus that observed in the “conventional ” heavy traffic regime. The latter phenomenon is illustrated by means of a numerical example having two customer classes.

Abstract. On the line and its tensor products, Fekete points are known to be the Gauss–Lobatto quadrature points. But unlike high-order quadrature, Fekete points generalize to non-tensor-product domains such as the triangle. Thus Fekete points might serve as an alternative to the Gauss–Lobatto points for certain applications. In this work we present a new algorithm to compute Fekete points and give results up to degree 19 for the triangle. For degree d>10 these points have the smallest Lebesgue constant currently known. The computations validate a conjecture of Bos [J. Approx. Theory, 64 (1991), pp. 271–280] that Fekete points along the boundary of the triangle are the one-dimensional Gauss–Lobatto points.

This article describes Archimedes, an automated system for solving partial differential equations on geometrically complex domains using distributed memory supercomputers. The tasks of such a system are manifold. First, Archimedes discretizes the object or domain being modeled by generating an unstructured mesh that fills the region. Then, the domain is partitioned into separate subdomains, which are mapped onto individual processors. Communication is routed between these processors. Finally, code is generated to solve a PDE in parallel. We shall discuss how we have automated each of these tasks, and describe our efforts to bring state-of-the-art computing to state-of-the-art engineering. Figure 4: Electric guitar. Figure 5: Finite element mesh. Finite Element Methods The most general numerical techniques for solving PDEs are known as finite element methods (FEM). This section describes the structure of the equations that result upon application of FEM. Readers wishing a proper treatment of FEM should consult standard texts such as Becker, Carey, and Oden [2] for an introduction, and Strang and Fix [19] for mathematical analysis. Consider a heat conduction problem posed on the two-dimensional domain of Figure 4, an electric guitar. The problem is to find the steady state temperature u(x; y) of the guitar, given that the guitar is exposed to specified heat sources. The physical behavior of this system is modeled by a partial differential equation

Scientific and engineering database applications put new requirements on database management systems that is usually not associated with traditional administrative database applications. These new database applications include finite element analysis (FEA) for computational mechanics and usually have a high level of complexity of both data and algorithms, as well as high volume of data and high requirements on execution efficiency. This paper shows how next generation object-oriented database technology that includes a relationally complete and extensible object-oriented query language can be used to model and manage FEA. The technology allows the design of domain models that represent application-oriented conceptual models of data and operators. An initial integration of a main-memory object-relational database management system with a state-of-the-art FEA program is presented. The FEA program integrates the complete FEA process and is controlled completely through a graphical user in...

This paper proposes an algorithm, referred to as BNA/FM (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNA/FM algorithm is based on finite element method and an extension of a generic algorithm developed by Dai and Harrison (1991). It uses piecewise polynomials to form an approximate subspace of an infinite dimensional functional space. The BNA/FM algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to BNA/SM (Brownian network analyzer with spectral method) of Dai and Harrison (1991), where global polynomials are used to form the approximate subspace and it sometime fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported that may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers are presented to illustrate the effectiveness of the Brownian approximation model and our BNA/FM algorithm.

. A finite-element solution of the time-dependent mass-continuity equation for column-averaged ice sheet flow and sliding is applied to the Antarctic Ice Sheet. First a calibration of the model to the steady-state present ice sheet configuration is presented. With fitted values of the parameters describing the regions of sliding, the degree of bed coupling, and the ice hardness, a change in the mean annual sea-level temperature is used to simulate variation of the climatic conditions over Antarctica for both warming and cooling of the climate. Paradoxically a climate warming of up to 9 degrees leads to an increase in ice volume, while cooling leads to decreasing ice volume as long as the present margins of Antarctica are maintained. Some extreme simulations of the Antarctic Ice Sheet for "maximum over-riding" and "minimum warm-climate" are shown for situations where the present bed conditions are altered. Finally a time-dependent simulation shows the response of the ice sheet system to...

This study investigates the existence and stability of static liquid bridges in non-axisymmetrically buckled elastic tubes. The liquid bridge which occludes the tube is formed by two menisci which meet the tube wall at a given contact angle along a contact line whose position is initially unknown. Geometrically non-linear shell theory is used to describe the deformation of the linearly elastic tube wall in response to an external pressure and to the loads due to the surface tension of the liquid bridge. This highly non-linear problem is solved numerically by Finite Element methods. It is found that for a large range of parameters (surface tension, contact angle and external pressure), the compressive forces generated by the liquid bridge are strong enough to hold the tube in a buckled configuration. Typical meniscus shapes in strongly collapsed tubes are shown and the stability of these configurations to quasi-steady perturbations is examined. The minimum volume of fluid requ...

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We propose a new spectral element method based on Fekete points. We use the Fekete criterion to compute points which are almost optimal for approximation, differentiation and quadrature. These grids have the same number of points as the dimension of the associated polynomial space, thus allowing us to use a cardinal function basis which leads to a diagonal mass matrix. For quadrilaterals, Gauss-Lobatto points are the unique Fekete points, making the method equivalent to the standard spectral element method. But unlike the Gauss-Lobatto points, Fekete points generalize to other domains such as the triangle. Furthermore, numerical and theoretical evidence suggests that the element boundary points are the Gauss-Lobatto points, making Fekete point triangular elements and quadrilateral elements naturally conform. Thus triangles and quadrilaterals can be combined in the same grid while retaining a diagonal mass matrix. We present an algorithm to compute Fekete points along with results for...