Numerical Benchmark Study Research Articles

A solar mean field dynamo benchmark

Context. The solar magnetic activity and cycle are linked to an internal dynamo. Numerical simulations are an e cient and accurate tool to investigate such intricate dynamical processes. Aims. We present the results of an international numerical benchmark study based on two-dimensional axisymmetric mean field solar dynamo models in spherical geometry. The purpose of this work is to provide the scientific community with reference cases that can be analyzed in detail and that can help in further development and validation of numerical codes that solve such problems. Methods. The results of eight numerical codes solving the induction equation in the framework of mean field theory are compared for three increasingly computationally intensive models of the solar dynamo: an dynamo with constant magnetic di usivity, an dynamo with magnetic di usivity sharply varying with depth and an example of a flux-transport Babcock-Leighton dynamo which includes a non-local source term and one large single cell of meridional circulation per hemisphere. All cases include a realistic profile of di erential rotation and thus a sharp tachocline. Results. The most important finding of this study is that all codes agree quantitatively to within less than a percent for the dynamo cases and within a few percents for the flux-transport case. Both the critical dynamo numbers for the onset of dynamo action and the corresponding cycle periods are reasonably well recovered by all codes. Detailed comparisons of butterfly diagrams and specific cuts of both toroidal and poloidal fields at given latitude and radius confirm the good quantitative agreement. Conclusions. We believe that such a benchmark study will be a very useful tool for the scientific community since it provides detailed standard cases for comparison and reference.

Crack blunting and the strength of soft elastic solids

When a material is so soft that the cohesive strength (or adhesive strength, in the case of interfacial fracture) exceeds the elastic modulus of the material, we show that a crack will blunt instead of propagating. Large–deformation finite–element model (FEM) simulations of crack initiation, in which the debonding processes are quantified using a cohesive zone model, are used to support this hypothesis. An approximate analytic solution, which agrees well with the FEM simulation, gives additional insight into the blunting process. The consequence of this result on the strength of soft, rubbery materials is the main topic of this paper. We propose two mechanisms by which crack growth can occur in such blunted regions. We have also performed experiments on two different elastomers to demonstrate elastic blunting. In one system, we present some details on a void growth mechanism for ultimate failure, post–blunting. Finally, we demonstrate how crack blunting can shed light on some long–standing problems in the area of adhesion and fracture of elastomers.

SUMMARY OF BENCHMARK NUMERICAL STUDIES FOR 3-D NATURAL CONVECTION IN AN AIR-FILLED ENCLOSURE

A numerical benchmark study dealing with 3-D natural convection in an air-filled cavity oriented at three different angles was held at the International Center for Heat and Mass Transfer (ICHMT) 2nd International Symposium on Advances in Computational Heat Transfer (CHT'01), May 20 - 25, 2001, in Palm Cove, Australia. The benchmark problem derives from an experimental study conducted by Leong, Hollands, and Brung involving natural convection within a differentially heated cubical cavity oriented at three different angles (φ) and subjected to four different Rayleigh numbers ( Ra ), all in the range 10 5 h Ra h 10 8 . The cube sidewall temperature varied linearly with distance from the hot face. In view of the number of test cases reported and the relatively low experimental uncertainty (about 1% of measurement), the problem appeared to be well suited as a benchmark for numerical validation. Ten teams (sets of participants) submitted papers using various mesh sizes and computational techniques. Most teams reported the existence of oscillatory solutions for cases at high Ra where the heating was primarily from below. Computed Nusselt numbers agreed with measurements at Ra and at inclinations where the heating was at least partly from the side. However, no one set of participants produced Nusselt numbers that agreed with the experimental results for all test cases.

Stabilized Finite Element Formulations for Shells in a Bending Dominated State

We consider the design of finite element methods for the Naghdi shell model in the case when the deformation is bending dominated. Two formulations based on stabilizing techniques are introduced and it is proved that they are stable, hence free from locking. The theoretical estimates are confirmed by numerical benchmark studies.