3D Partitions Research Articles

Investigation of Sandwich Panel Under High Temperature Loading Using Finite Element Analysis

Structural and nonstructural members are always encountered with extreme loading conditions. In case of nonstructural members, the failure happens much sooner due to lack of load capacity. The nonstructural elements are not designed for structural loading conditions. However, they might be impacted to other loading types such as fire. In recent years, the use of 3D panel is one of the most common partitions in the world. In this research, the effect of fire on 3D panel partitions is presented. To achieve this goal, ABAQUS finite element software is employed to simulate the 3D panel. The 3D panel is embedded inside steel frame. In order to have an accurate result, two validations are performed including steel frame and the steel frame with 3D panel. In the study, 5 different fire loadings 200, 400, 600, 800 and 1000°C are applied into the model. Results include load-displacement diagram and stiffness and ductility factors. The results indicated that as fire load increases, the stiffness and ductility factor are reduced.

Near-isotropic noncontrast MRA of the renal and peripheral arteries using a thin-slab stack-of-stars quiescent interval slice-selective acquisition.

Noncontrast MRA avoids potential risks from gadolinium-based contrast agents. A 2D noncontrast technique, quiescent interval slice-selective (QISS), accurately evaluates the peripheral arteries but has limited spatial resolution along the slice direction. We therefore implemented a prototype thin-slab stack-of-stars version (tsSOS-QISS) with nearly isotropic spatial resolution and tested it in the renal and peripheral arteries of healthy subjects and patients with vascular disease. The study was approved by the hospital institutional review board. A total of 16 subjects were scanned at 1.5 T: 7 for imaging of the renal arteries and 9 for imaging of the peripheral arteries. For tsSOS-QISS of the renal arteries, each slab consisted of about sixteen 1.3-mm-thick or 2.0-mm-thick slices (interpolated to thirty-two 0.65-mm-thick or 1.0-mm-thick 3D partitions) oriented in an oblique axial or oblique coronal view along the length of the target vessel and was acquired in a breath-hold. For tsSOS-QISS of the peripheral arteries, 20 axial overlapping thin slabs were typically acquired, each with twelve 1.3-mm-thick slices (interpolated to twenty-four 0.65-mm-thick 3D partitions). Image quality, vessel sharpness in multiplanar reconstructions, and normalized SNR were measured. Image quality and normalized SNR in the renal and peripheral arteries were significantly better compared with 2D QISS acquired at the same spatial resolution, while vessel sharpness was improved in multiplanar reconstructions of the renal arteries. The tsSOS-QISS technique overcomes a significant limitation of 2D QISS by providing nearly isotropic spatial resolution with improved image quality, normalized SNR, and vessel sharpness in multiplanar reconstructions.

Trace identities for the topological vertex

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson-Thomas invariants of elliptically fibered Calabi-Yau threefolds at the end of the paper.

The arc space of the Grassmannian

We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.

Mixed-Dimensional Coupling for Parallel Partitioned Nonlinear Finite-Element Analysis

AbstractThis paper presents a novel mixed-dimensional coupling method for parallel partitioned analysis, allowing efficient nonlinear structural analysis with selective deployment of differently dimensioned elements. With this method, a structural domain is decomposed into partitions that are processed in parallel using a recently developed dual super-element partitioning approach. Individual partitions consist exclusively of reduced dimensioned (1D) or higher dimensioned (3D continuum) elements, and coupling between 1D and 3D partitions is introduced at the boundary of 3D partitions, enabling a reduction in the communication overhead. Several examples are provided to verify the proposed approach and demonstrate its benefits for nonlinear structural analysis.

A Gaussian Distribution for Refined DT Invariants and 3D Partitions

We show that the refined Donaldson-Thomas invariants of C3, suitably normalized, have a Gaussian distribution as limit law. Combinatorially these numbers are given by weighted counts of 3D partitions. Our technique is to use the Hardy-Littlewood circle method to analyze the bivariate asymptotics of a q-deformation of MacMahon's function. The proof is based on that of E.M. Wright who explored the single variable case.

Methodology for automatic recovering of 3D partitions from unstitched faces of non-manifold CAD models

Data exchanges between different software are currently used in industry to speed up the preparation of digital prototypes for finite element analysis (FEA). Unfortunately, due to data loss, the yield of the transfer of manifold models rarely reaches 1. In the case of non-manifold models, the transfer results are even less satisfactory. This is particularly true for partitioned 3D models: during the data transfer based on the well-known exchange formats, all 3D partitions are generally lost. Partitions are mainly used for preparing mesh models required for advanced FEA: mapped meshing, material separation, definition of specific boundary conditions, etc. This paper sets up a methodology to automatically recover 3D partitions from exported non-manifold CAD models to increase the yield of the data exchange. Our fully automatic approach is based on three steps. First, starting from a set of potentially disconnected faces, the CAD model is stitched. Then, the shells used to create the 3D partitions are recovered using an iterative propagation strategy which starts from the so-called manifold vertices. Finally, using the identified closed shells, the 3D partitions can be reconstructed. The proposed methodology has been validated on academic as well as industrial examples.

The orbifold topological vertex

We develop a topological vertex formalism for computing the Donaldson–Thomas invariants of Calabi–Yau orbifolds. The basic combinatorial object is the orbifold vertex V λ μ ν G , a generating function for the number of 3D partitions asymptotic to 2D partitions λ, μ, ν and colored by representations of a finite Abelian group G acting on C 3 . In the case where G ≅ Z n acting on C 3 with transverse A n − 1 quotient singularities, we give an explicit formula for V λ μ ν G in terms of Schur functions. We discuss applications of our formalism to the Donaldson–Thomas crepant resolution conjecture and to the orbifold Donaldson–Thomas/Gromov–Witten correspondence. We also explicitly compute the Donaldson–Thomas partition function for some simple orbifold geometries: the local football P a , b 1 and the local B Z 2 gerbe.

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.

Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems

In this paper, the fifth in a series, the high-order spectral finite-volume, or spectral volume (SV) method for unstructured grids is extended to three dimensions. Limitations of conventional structured and unstructured methods are first reviewed. The spectral finite-volume method for generalized conservation laws is then described. It is shown that if all grid cells are partitioned into structured sub-cells in a similar manner, the discretizations become universal, and are reduced to the same weighted sum of unknowns involving just a few simple adds and multiplies. Important aspects of the data structure and its effects on communication and the optimum use of cache memory are discussed. Previously defined one-parameter partitions of the SV in 2D are extended to multiple parameters and then used to construct 3D partitions. Numerical solutions of plane electromagnetic waves incident on perfectly conducting circular cylinders and spheres are presented and compared with the exact solution to demonstrate the capability of the method. Excellent agreement has been found. Computation timings show that the new method is more efficient than conventional structured and unstructured methods.

Building 3D frameworks of accessory aquatic systems

We consider a mathematical framework for spatially describing accessory aquatic systems that belong to a major hydric complex like a large reservoir created by damming a river. The central purpose is to provide a precise and complete mathematical covering of short-scale or localized water bodies, with typical lengths between 1 to 10 km, that are laterally attached to the main river. Such bodies may deserve a more detailed mathematical representation due, for instance, to their tendency to develop stagnant like hydric behaviors. The proposed framework may work as an infrastructure for developing and/or installing dynamic water quality models. It can generate tetrahedrizations of the aquatic system in question by working according to a procedure which builds, successively, reliable candidates for the following entities: (a) Free Surface Triangular Mesh; (b) Submerged Terrain Triangular Mesh; (c) Three Dimensional Partition of the Domain; (d) Basic Tetrahedrization of 3D Partitions; and (e) Refinement of the Basic Tetrahedrization through a Multi-Layer Tetrahedrization Algorithm. The required input for this procedure is only composed by terrain contour data and 3D located points. We present example applications including a real scenario belonging to a recent flooded system in Brazil.

Parallel multigrid finite volume computation of three-dimensional thermal convection

A parallel implementation of the finite volume method for three-dimensional, time-dependent, thermal convective flows is presented. The algebraic equations resulting from the finite volume discretization, including a pressure equation which consumes most of the computation time, are solved by a parallel multigrid method. A flexible parallel code has been implemented on the Intel Paragon, the Cray T3D, and the IBM SP2 by using domain decomposition techniques and the MPI communication software. The code can use 1D, 2D, or 3D partitions as required by different geometries, and is easily ported to other parallel systems. Numerical solutions for air (Prandtl number Pr = 0.733) with various Rayleigh numbers up to 10 7 are discussed.