Nodal Geometry Research Articles

Development and benchmarking of SIMULATE5-K-VVER

SIMULATE5-K-VVER (S5K-VVER) is the newest addition to the CMS5-VVER modeling suite developed by Studsvik Scandpower, Inc. (SSP). In S5K-VVER, the coupled time-dependent multigroup neutron diffusion equation, delayed neutron precursor concentrations, and thermal hydraulic models are solved in triangular nodal geometry to model transients in Water-Water Energetic Reactors (VVERs).The neutron kinetics equations in S5K-VVER are solved in a two-step process: temporal integration and spatial integration. In S5K-VVER there are two options for temporal integration: the Frequency Transform Method (FTM) and a Backward-Euler Method (BEM).S5K-VVER is then used to solve a series of benchmark problems including the AER-DYN-001 and AER-DYN-002 numerical transient benchmarks. Finally, the Kalinin-3 benchmark is modeled which describes the shutting-off of a single Main Coolant Pump (MCP) in the Kalinin-3 VVER-1000/320 nuclear power plant. Compared to plant data, the S5K-VVER solution agrees within 1.5% of core total power during this transient.

Nodal deficiency of random spherical harmonics in presence of boundary

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere, satisfying the Dirichlet boundary conditions along the equator. For this model, we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relative to the rotation invariant model of random spherical harmonics. Jean Bourgain’s research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations {Theorem 2.2 in the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain’s work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

Image-Based Geometrical Characterization of Nodes in Additively Manufactured Lattice Structures.

Additive manufacturing (AM) enables the fabrication of lattice structures with optimal mechanical, fluid, and thermal properties. However, during the AM fabrication process, defects are produced in the strut and node elements, which comprise the lattice structure. This leads to discrepancies between the AM fabricated lattice and its idealized computer-aided design (CAD) model, negatively affecting the ability to predict the mechanical behavior of the fabricated lattice via numerical models. Current research is focused on quantification of geometric uncertainties in the strut elements of the lattice; as-manufactured node geometries remain relatively unexplored on an individual scale, despite their criticality to the mechanical response of the structure. Understanding the geometrical properties of as-manufactured nodes relative to CAD idealizations can be used to improve lattice designs and numerical models. In this research, X-ray microcomputed tomography (μCT) is used to analyze and quantify the as-manufactured nodal geometry, found in face-centered cubic and face-centered cubic with axial struts lattices fabricated via selective laser melting. A custom tool is developed that enables auto-isolation and classification of nodal joints from μCT-derived cross-sectional slices. Geometrical properties are extracted from the isolated nodal cross sections and compared with their idealized CAD model counterpart. Quantification of geometrical defects provides insight into how nodes within an AM lattice structure differ from each other and their idealized design. Overall, this research is an initial step toward developing accurate and efficient numerical models, as well as better node design for AM.

Some remarks on nodal geometry in the smooth setting

We consider a Laplace eigenfunction $$\varphi _\lambda $$ on a smooth closed Riemannian manifold, that is, satisfying $$-\Delta \varphi _\lambda = \lambda \varphi _\lambda $$ . We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $$\varphi _\lambda $$ . This extends previous work of Jakobson and Mangoubi in case (M, g) is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. Ann Math (2) 187(1):241–262, 2018). Second, we exhibit some remarks related to the asymptotic geometry of nodal domains. In particular, we observe an analogue of a result of Cheng in higher dimensions regarding the interior opening angle of a nodal domain at a singular point. Further, for nodal domains $$\Omega _\lambda $$ on which $$\varphi _\lambda $$ satisfies exponentially small $$L^\infty $$ bounds, we give some quantitative estimates for radii of inscribed balls.

Inverse form-finding for tensegrity structures

In this study we examine the topic of inverse form-finding, also referred to as topology finding, for tensegrity structures. Specifically, the problem addressed is given the specification of final nodal positions for a tensegrity, we seek to find appropriate connectivities, or topologies, that satisfy stability and connectivity constraints. Two new algorithms are presented in the paper. The first may be applied for generating prestress-stable tensegrity structures, while the second can be used to generate super-stable tensegrities. Numerical examples for both 2D and 3D tensegrities are provided to demonstrate that these new algorithms can produce desirable structures with nodal positions being the only prescribed piece of information. We also show that inverse form-finding of a specific Class k tensegrity can be formulated into a graph factorization problem. This is the first time that both the stability property and class can be specified among the few existing inverse form-finding methods. These new methods facilitate the design process in which a desired nodal geometry is prescribed, and viable structural configurations consistent with this geometry can be obtained.

Nodal geometry, heat diffusion and Brownian motion

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $\Omega_\lambda$ almost fully contains a ball of radius $\sim \frac{1}{\sqrt{\lambda}}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.

Spectral geometry of the Steklov problem (Survey article)

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.

SMITHERS: An object-oriented modular mapping methodology for MCNP-based neutronic–thermal hydraulic multiphysics

A novel object-oriented modular mapping methodology for externally coupled neutronics–thermal hydraulics multiphysics simulations was developed. The Simulator using MCNP with Integrated Thermal-Hydraulics for Exploratory Reactor Studies (SMITHERS) code performs on-the-fly mapping of material-wise power distribution tallies implemented by MCNP-based neutron transport/depletion solvers for use in estimating coolant temperature and density distributions with a separate thermal-hydraulic solver. The key development of SMITHERS is that it reconstructs the hierarchical geometry structure of the material-wise power generation tallies from the depletion solver automatically, with only a modicum of additional information required from the user. Additionally, it performs the basis mapping from the combinatorial geometry of the depletion solver to the required geometry of the thermal-hydraulic solver in a generalizable manner, such that it can transparently accommodate varying levels of thermal-hydraulic solver geometric fidelity, from the nodal geometry of multi-channel analysis solvers to the pin-cell level of discretization for sub-channel analysis solvers.The mapping methodology was specifically developed to be flexible enough such that it could successfully integrate preexisting depletion solver case files with different thermal-hydraulic solvers. This approach allows the user to tailor the selection of a thermal-hydraulic solver to the requirements and limitations of the specific problem under consideration, without needing to modify their existing depletion code input files. To enable support of a wide range of existing depletion solver input decks, SMITHERS can accommodate arbitrarily detailed geometry segmentation for the depletion calculation with a minimum of additional information required from the user. This new implementation was motivated by the desire to enable easier multiphysics modeling of a wide range of reactor types, from power reactors to research reactors.The initial implementation of this modular mapping methodology incorporates the neutronic depletion solver MCNP/MONTEBURNS and a multi-channel analysis two-phase flow thermal hydraulic solver. Two verification test problems were evaluated to verify that the code’s routines were operating as intended. Preexisting generic BWR and nontraditional PWR cases were selected to ensure maximum code coverage and evaluate the operation of all implemented mapping routines. The obtained power, coolant temperature, and coolant density results verified that SMITHERS was correctly performing on-the-fly mapping combinatorial-basis MCNP/MONTEBURNS-calculated material powers to the thermal-hydraulic solver’s nodal geometry and that nodal coolant temperature and densities were correctly returned to the combinatorial geometry of MCNP.

Nodal geometry of graphs on surfaces

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the $n$th Laplacian eigenvalue is at most $2\left[ 6(n-1) + 15(2g-2) \right]^2$. Our results hold for any Schr\"odinger operator, not just the Laplacian.

Modeling spinal cord contusion, dislocation, and distraction: Characterization of vertebral clamps, injury severities, and node of Ranvier deformations

Spinal cord contusion and transection models are widely used for studying spinal cord injury (SCI). Clinically, however, other biomechanical injury mechanisms such as vertebral dislocation and distraction frequently occur, but these injuries are difficult to produce in animals. We mechanically characterize a vertebral clamping strategy that enables the modeling of vertebral dislocation and distraction injuries – in addition to the standard contusion paradigm – in the rat cervical spine. These vertebral clamps have a stiffness of 83.6±18.9N/mm and clamping strength 64.7±10.2N which allows injuries to be modeled at high-speed (∼100cm/s). Logistic regression indicated that a moderate-to-severe injury, with an acute mortality rate of 10%, occurs at 2.6mm of C4/5 dorso-ventral dislocation and 4.1mm of rostro-caudal distraction between C4 and C5. Injuries produced by dislocation and distraction exhibited features of axonal damage that were absent in contusion injuries. We conducted morphometric analysis at the nodes of Ranvier using immunohistochemistry for potassium channels (Kv1.2) in the juxtaparanodal region. Following distraction injuries, elongated nodes of Ranvier were observed up to 4mm rostral to the lesion. In contrast, contusion injuries produced distortions in nodal geometry which were restricted to the vicinity of the lesion. The greatest deformations in node of Ranvier geometry occurred at the dislocation epicenter. Given the importance of white matter damage in SCI pathology, the distinctiveness of these injury patterns demonstrate that the dislocation and distraction injury models complement existing contusion models. Together, these three animal models span a broader clinical spectrum for more reliably gauging the potential human efficacy of therapeutic strategies.

Sign and area in nodal geometry of Laplace eigenfunctions

The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign( f ) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line { f = 0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mild local asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and Nadirashvili with a new result on harmonic functions in the unit disc.

Covering lemmas and an application to nodal geometry on Riemannian manifolds

The main part of this note is to show a general covering lemma in Rn, n 2 2 , with the aim to obtain the estimate for BMO norm and the volume of a nodal set of eigenfunctions on Riemannian manifolds. This article is a continuation of our previous work [L]. In [L] we proved a covering lemma in R2 and applied it to the BMO norm estimates for eigenfunctions on Riemannian surfaces. The principal part of this article is to prove a general covering lemma in Rn for n 2 2 . As applications, we can obtain the BMO estimate for eigenfunctions and the volume estimate for the nodal set. Let M n be a smooth, compact, and connected Riemannian manifold with no boundary. Let A denote the Laplacian on M n . Let -Au = ilu, u an eigenfunction with eigenvalue il , il > 1 . Our main results can be stated as follows Theorem A (BMO estimate for log lul) . For u , il as above and n 2 3 , where C is independent of il and u and is only dependent on n and M n Theorem B (geometry of nodal domains). Let n 2 3 and u , il as above. Let B c M n be any ball, and let R c B be any of the connected components of {x E B :u(x) # 0) . If R intersects the middle halfof B , then where C is independent of il and u Donnelly and Fefferman [DFl, DF2] and Chanillo and Muckenhoupt [CM] proved Theorem A with (10gil)~ replaced by iln(n+2)/4 and iln log2 , respectively, and Theorem B with A-2n2-n14 replaced by il-(n+n2(n+2))/2 and il-2n2-n/2 (log il)-2n , respectively. Received by the editors June 28, 1991. 1991 Mathematics Subject Class$cation. Primary 35B05, 58G03. @ 1993 American Mathematical Society 0002-9939193 $1.00 + $.25 per page

Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds

Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds

Nodal geometry on Riemannian manifolds

Nodal geometry on Riemannian manifolds

Postnatal differentiation of rat optic nerve fibers: electron microscopic observations on the development of nodes of Ranvier and axoglial relations.

The postnatal differentiation of rat optic nerve fibres was examined by transmission electron microscopy. The results show that many early developing axons contain clusters of vesiculotubular profiles prior to myelination. At places vesicular elements appear to fuse with the axolemma, and, in addition, some axons exhibit deep axolemmal invaginations and axoplasmic lamellated bodies. It is suggested that these features might reflect axolemmal remodelling, possibly involving axoglial signalling and/or functional differentiation of the axolemma. The size distribution of unmyelinated optic nerve axons changes little during development. Ensheathment of larger axons commences 6 days postnatally. The subsequent formation of compact myelin sheaths is accompanied by an increase in axonal diameter. The early sheaths are a few microns long and separated by long bare axon segments. In optic nerves from 10-12-day-old rat pups, a few sheaths consisting of about five layers border primitive asymmetric nodes with a patchy axolemmal undercoating. Extensions from one of the terminating sheaths are often associated with undercoated patches of axolemma. Relatively differentiated nodes of Ranvier first appear 14-16 days after birth. The continued nodal maturation involves establishment of a regular nodal geometry, increasing distinctness of the axolemmal undercoating, and formation of perinodal astrocytic processes embedded in an extracellular node gap substance. The results are compared with available data on the conduction properties of rat optic nerve fibres during development.