Negative Jacobian Research Articles

Shilnikov attractors in three-dimensional orientation-reversing maps.

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing and orientation-preserving cases. We consider the three-dimensional Mirá map x¯=y,y¯=z,andz¯=Bx+Cy+Az-y2 with the negative Jacobian (B<0) as a basic model demonstrating various types of Shilnikov attractors. We show that depending on values of parameters A,B, and C, such attractors can be of three possible types: hyperchaotic (with two positive and one negative Lyapunov exponent), flow-like (with one positive, one very close to zero, and one negative Lyapunov exponent), and strongly dissipative (with one positive and two negative Lyapunov exponents). We study scenarios of the formation of such attractors in one-parameter families.

Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian

We construct an a.e. approximately differentiable homeomorphism of a unit $n$-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a.e. Moreover, the homeomorphism together with its inverse satisfy a rather general sub-Lipschitz condition, in particular it can be bi-Holder continuous with an arbitrary exponent less than $1$.

Robust structure simplification for hex re-meshing

We introduce a robust and automatic algorithm to simplify the structure and reduce the singularities of a hexahedral mesh. Our algorithm interleaves simplification operations to collapse sheets and chords of the base complex of the input mesh with a geometric optimization, which improves the elements quality. All our operations are guaranteed not to introduce elements with negative Jacobians, ensuring that our algorithm always produces valid hex-meshes, and not to increase the Hausdorff distance from the original shape more than a user-defined threshold, ensuring a faithful approximation of the input geometry. Our algorithm can improve meshes produced with any existing hexahedral meshing algorithm --- we demonstrate its effectiveness by processing a dataset of 194 hex-meshes created with octree-based, polycube-based, and field-aligned methods.

On the numerical modeling of granular material flows via the Particle Finite Element Method (PFEM)

The aim of this work is to describe a numerical framework for reliably and robustly simulating the different kinematic conditions exhibited by granular materials while spreading – from a stagnant condition, when the material is at rest, to a transition to granular flow, and back to a deposit profile. The gist of the employed modeling approach was already presented by the authors in a recent work (Cante et al., 2014), but no proper description of the underlying numerical techniques was provided therein. The present paper focuses precisely on the detailed discussion of such numerical techniques, as well as on its rigorous validation with the experimental results obtained by Lajeunesse et al. (2004).The constitutive model is based on the concepts of large strains plasticity. The yield surface is defined in terms of the Drucker Prager yield function, endowed with a deviatoric plastic flow and the elastic part by a hypoelastic model. The plastic flow condition is assumed nearly incompressible, so a u−p mixed formulation, with a stabilization of the pressure term via the Polynomial Pressure Projection (PPP), is employed. The numerical scheme takes as starting point the Particle Finite Element Method (PFEM) in which the spatial domain is continuously redefined by a different nodal reconnection, generated by a Delaunay triangulation. In contrast to classical PFEM approximations (Idelsohn et al., 2004), in which the free boundary is obtained by a geometrical technique (α-shape method), in this work the boundary is treated as a material surface, and the boundary nodes are removed or inserted by means of an error function. One of the novelties of this work is the use of the so-called Impl–Ex hybrid integration technique to enhance the spectral properties of the algorithmic tangent moduli and thus reduce the number of iterations and robustness of the accompanying Newton–Raphson solution algorithm (compared with fully implicit schemes respectively). The new set of numerical tools implemented in the PFEM algorithm – including new discretization techniques, the use of a projection of the variables between meshes, and the constraint of the free-surface instead using classic α-shape – allows us to eliminate the negative Jacobians present during large deformation problems, which is one of the drawbacks in the simulation of granular flows.Finally, numerical results are compared with the experiments developed in Lajeunesse et al. (2004), where a granular mass, initially confined in a cylindrical container, is suddenly allowed to spread by the sudden removal of the container. The study is carried out using different geometries with varying initial aspect ratios. The excellent agreement between computed and experimental results convincingly demonstrates the reliability of the model to reproduce different kinematic conditions in transient and stationary regimes.

Solid T-spline construction from boundary representations for genus-zero geometry

This paper describes a novel method to construct solid rational T-splines for complex genus-zero geometry from boundary surface triangulations. We first build a parametric mapping between the triangulation and the boundary of the parametric domain, a unit cube. After that we adaptively subdivide the cube using an octree subdivision, project the boundary nodes onto the input triangle mesh, and at the same time relocate the interior nodes via mesh smoothing. This process continues until the surface approximation error is less than a pre-defined threshold. T-mesh is then obtained by pillowing the subdivision result one layer on the boundary and its quality is improved. Templates are implemented to handle extraordinary nodes and partial extraordinary nodes in order to get a gap-free T-mesh. The obtained solid T-spline is C2-continuous except for the local region around each extraordinary node and partial extraordinary node. The boundary surface of the solid T-spline is C2-continuous everywhere except for the local region around the eight nodes corresponding to the eight corners of the parametric cube. Finally, a Bézier extraction technique is used to facilitate T-spline based isogeometric analysis. The obtained Bézier mesh is analysis-suitable with no negative Jacobians. Several examples are presented in this paper to show the robustness of the algorithm.

An electric charge has no screw sense – a comment on the twistfree formulation of electrodynamics by da Rocha and Rodrigues

AbstractDa Rocha and Rodigues (RR) claim (i) that in classical electrodynamics in vector calculus the distinction between polar and axial vectors and in exterior calculus between twisted and untwisted forms is inappropriate and superfluous, and (ii) that they can derive the Lorentz force equation from Maxwell's equations. As to (i), we point out that the distinction of polar/axial and twisted/untwisted derives from the property of the electric charge of being a pure scalar, that is, not carrying any screw sense. Therefore, the mentioned distinctions are necessary ingredients in any fundamental theory of electrodynamics. If one restricted the allowed coordinate transformations to those with positive Jacobian determinants (or prescribed an equivalent constraint), then the RR scheme could be accommodated; however, such a restriction is illegal since electrodynamics is, in fact, also covariant under transformations with negative Jacobians. As to (ii), the “derivation” of the Lorentz force from Maxwell's equations, we point out that RR forgot to give the symbol F (the field strength) in Maxwell's equations an operational meaning in the first place. Thus, their proof is empty. Summing up: the approach of RR does not bring in any new insight into the structure of electrodynamics.

An electric charge has no screw sense - a comment on the twistfree formulation of electrodynamics by da Rocha and Rodrigues

Da Rocha and Rodigues (RR) claim (i) that in classical electrodynamics in vector calculus the distinction between polar and axial vectors and in exterior calculus between twisted and untwisted forms is inappropriate and superfluous, and (ii) that they can derive the Lorentz force equation from Maxwell's equations. As to (i), we point out that the distinction of polar/axial and twisted/untwisted derives from the property of the electric charge of being a pure scalar, that is, not carrying any screw sense. Therefore, the mentioned distinctions are necessary ingredients in any fundamental theory of electrodynamics. If one restricted the allowed coordinate transformations to those with positive Jacobian determinants (or prescribed an equivalent constraint), then the RR scheme could be accommodated; however, such a restriction is illegal since electrodynamics is, in fact, also covariant under transformations with negative Jacobians. As to (ii), the "derivation" of the Lorentz force from Maxwell's equations, we point out that RR forgot to give the symbol F (the field strength) in Maxwell's equations an operational meaning in the first place. Thus, their proof is empty. Summing up: the approach of RR does not bring in any new insight into the structure of electrodynamics.

Clustered generalized finite element methods for mesh unrefinement, non‐matching and invalid meshes

AbstractIn spite of significant advancements in automatic mesh generation during the past decade, the construction of quality finite element discretizations on complex three‐dimensional domains is still a difficult and time demanding task.In this paper, the partition of unity framework used in the generalized finite element method (GFEM) is exploited to create a very robust and flexible method capable of using meshes that are unacceptable for the finite element method, while retaining its accuracy and computational efficiency. This is accomplished not by changing the mesh but instead by clustering groups of nodes and elements. The clusters define a modified finite element partition of unity that is constant over part of the clusters. This so‐called clustered partition of unity is then enriched to the desired order using the framework of the GFEM.The proposed generalized finite element method can correctly and efficiently deal with: (i) elements with negative Jacobian; (ii) excessively fine meshes created by automatic mesh generators; (iii) meshes consisting of several sub‐domains with non‐matching interfaces. Under such relaxed requirements for an acceptable mesh, and for correctly defined geometries, today's automated tetrahedral mesh generators can practically guarantee successful volume meshing that can be entirely hidden from the user. A detailed technical discussion of the proposed generalized finite element method with clustering along with numerical experiments and some implementation details are presented. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Symbolic analysis of attractor geometry for the Lozi map.

Symbolic dynamics of the Lozi map is described for both positive and negative Jacobians. Based on the ordering rules of symbolic sequences, the geometrical structure of foliations associated with attractors is constructed for some typical cases. The critical parameters between one- and two-piece attractors are also given.

A warning against misuse of quarter‐point elements

AbstractQuarter‐point elements are used very frequently for fracture mechanics computations, because the quarter‐point technique yields the required singular interpolation without any modification to existing software. This advantage is particularly significant for three‐dimensional stress intensity factor computations because of the difficulty of implementing other techniques. However, in practical 3‐D applications, the crack front is usually curved, and this note proves that a crack front distortion leads to a negative Jacobian in the region surrounding the crack front. The numerical difficulties to be expected depend on the aspect ratio of the elements.