Discontinuous Fields Research Articles

Generalized layerwise mechanics for the static and modal response of delaminated composite beams with active piezoelectric sensors

A coupled linear layerwise laminate theory and a beam FE are formulated for analyzing delaminated composite beams with piezoactuators and sensors. The model assumes zig-zag fields for the axial displacements and the electric potential and it treats the discontinuities in the displacement fields due to the delaminations as additional degrees of freedom. The formulation naturally includes the excitation of piezoelectric actuators, their interactions with the composite laminate, and the effect of delamination on the predicted sensory voltage. The quasistatic and modal response of laminated composite Gr/Epoxy beams with active or sensory layers having various delamination sizes is predicted. The numerical results illustrate the strong effect of delamination on the sensor voltage, on through the thickness displacement and on the stress fields. Finally, the effect of delamination on modal frequencies and shapes are predicted and compared with previously obtained experimental results.

On Poynting's Theorem and Reciprocity Relations for Discontinuous Fields

Poynting's theorem and reciprocity relations are investigated, based on the postulate that Maxwell's equations are always valid in the sense of distributions. Three special cases of practical importance in which the fields possess bounded discontinuities on simple (nonmaterial) and material interfaces between two different media are rigorously examined, incorporating the basic properties of the Heaviside unit step function and the Dirac delta distribution for multiple variables.

Interfacial Cracks of Antiplane Sliding Mode between Usual Elastic Materials and Quasicrystals

The present study deals with the problem of interfacial cracks of antiplane sliding mode between a usual elastic material and a one-dimensional hexagonal quasicrystal. Based on the physical facts that balance of the phason stress field is foreign to the real force in or out of quasicrystal in physical space, and the quasicrystal is bonded to a usual elastic material without both phason displacement and stress fields, the problem is described by analytic functions and attributed to find solutions of the Riemann-Hilbert problem. It is found that the stress intensity factor is not related to phason strain field, and the phason stress field does not exist. The discontinuity of phonon displacement field across crack is related to the phason displacement field because of the coupling of phonon and phason strain fields. Although there is not the phason displacement on the bonded portion of interface, it exists on the crack’s surface. The energy release rate obtained from interfacial crack’s propagating is different from that of an interfacial crack between two different pure elastic materials.

Extension of a V-notched bar and failure of plastic bodies

The effect of plastic strain localization near the domains of sharp variation in shape and transverse cross-section of bodies is well known. But such processes have not yet been studied analytically well enough. On the basis of the model of an ideally rigid-plastic body, we propose an approach for determining the strain fields near the concentrators on the basis the motion of the displacement velocity field (near surfaces or discontinuity lines in the form of rigid-plastic boundaries and centers of the fan of slip lines under plane strain). We consider the problem on plastic flow with failure for a V-notched bar. We show that the plastic flow is not unique (in the framework of the solution completeness).We propose to use the strain criterion for choosing the preferable plastic flow. On the basis of the solutions thus obtained, we state an approach to studying failure processes for more complicated models of bodies.

A unified variational approach for coupling 3D–1D models and its blood flow applications

In this work, we propose an extended variational formulation in order to handle the problem of the flow of an incompressible fluid in compliant vessels with discontinuous fields. This approach makes possible to tackle in a consistent variational manner problems like the coupling between models of different dimensionality. The main ideas of the work are presented for a simple heat transfer problem and then are used in the context of modelling blood flow in large vessels. A computational implementation of the proposed formulation is also shown and discussed by presenting examples of practical applications.

Global asymptotic stability for a class of discontinuous vector fields in

Consider a partition of the plane by squares of the same size. For the class of discontinuous vector fields formed by infinitely many linear vector fields on every square of the above partition, we provide sufficient conditions for the existence of a global asymptotic equilibrium point. Such vector fields come from control theory.

P-Adaptive C k generalized finite element method for arbitrary polygonal clouds

A p-Adaptive Generalized Finite Element Method (GFEM) based on a Partition of Unity (POU) of arbitrary smoothness degree is presented. The shape functions are built from the product of a Shepard POU and enrichment functions. Shepard functions have a smoothness degree directly related to the weighting functions adopted in their definition. Here the weighting functions are obtained from boolean R-functions which allow the construction of C k approximations, with k arbitrarily large, defined over a polygonal patch of elements, named cloud. The Element Residual Method is used to obtain error indicators by taking into account the typical nodal enrichment scheme of the method. This procedure is enhanced by using approximations with a high degree of smoothness as it eliminates the discontinuity of the stress field in the interior of each cloud. Adaptive analysis of plane elasticity problems are presented, and the performance of the technique is investigated.

Modeling and forecasting solar energetic particle events at Mars: the event on 6 March 1989

Context. Large solar energetic particle events are able to enhance the radiation intensity present in interplanetary space by several orders of magnitude. Therefore the study, modeling and prediction of these events is a key factor to understand our space environment and to protect manned space missions from hazardous radiation. Aims. We model an intense solar energetic particle event observed simultaneously on the 6 of March 1989 by the near-Earth orbiting spacecraft IMP-8 and by the Phobos-2 spacecraft in orbit around Mars (located 72° to the East of the Earth and at 1.58 AU from the Sun). This particle event was associated with the second largest X-ray flare in solar cycle 22. The site of this long-duration X15/3B solar flare was N35E69 (as seen from the Earth) and the onset of the 1-8 A X-ray emission occurred at 1350 UT on 6 March 1989. A traveling interplanetary shock accompanied with <15 MeV proton intensity enhancements was observed by IMP-8 at 1800 UT on 8 March and by Phobos-2 at 2015 UT on 9 March. This shock determines the particle intensities at both spacecraft. Methods. We use an MHD code to model the propagation of the associated shock to both spacecraft and a particle transport code to model the proton intensities measured by IMP-8 and Phobos-2. By assuming that energetic particles are continuously accelerated by the traveling shock, and that the injection rate of these particles, Q, into the interplanetary medium is related to the upstream-to-downstream velocity ratio, VR, at the point of the shock front that connects with the observer, we perform predictions of the solar energetic particle intensities observed at Mars from those measured at Earth. Results. We reproduce not only the arrival times of the shock at both spacecraft but also the measured jump discontinuity of solar wind speed, density and magnetic field. Also, we reproduce the 0.5-20 MeV proton intensities measured by both spacecraft. Functional dependences such as the Q(VR) relation deduced here allow us to predict the proton intensities measured at Phobos-2 for this event. Applications of this model for future predictions of solar energetic particle fluxes at Mars are discussed.

An alternative to the Mononobe–Okabe equations for seismic earth pressures

A closed-form stress plasticity solution is presented for gravitational and earthquake-induced earth pressures on retaining walls. The proposed solution is essentially an approximate yield-line approach, based on the theory of discontinuous stress fields, and takes into account the following parameters: (1) weight and friction angle of the soil material, (2) wall inclination, (3) backfill inclination, (4) wall roughness, (5) surcharge at soil surface, and (6) horizontal and vertical seismic acceleration. Both active and passive conditions are considered by means of different inclinations of the stress characteristics in the backfill. Results are presented in the form of dimensionless graphs and charts that elucidate the salient features of the problem. Comparisons with established numerical solutions, such as those of Chen and Sokolovskii, show satisfactory agreement (maximum error for active pressures about 10%). It is shown that the solution does not perfectly satisfy equilibrium at certain points in the medium, and hence cannot be classified in the context of limit analysis theorems. Nevertheless, extensive comparisons with rigorous numerical results indicate that the solution consistently overestimates active pressures and under-predicts the passive. Accordingly, it can be viewed as an approximate lower-bound solution, than a mere predictor of soil thrust. Compared to the Coulomb and Mononobe–Okabe equations, the proposed solution is simpler, more accurate (especially for passive pressures) and safe, as it overestimates active pressures and underestimates the passive. Contrary to the aforementioned solutions, the proposed solution is symmetric, as it can be expressed by a single equation—describing both active and passive pressures—using appropriate signs for friction angle and wall roughness.

Upper bound limit analysis using discontinuous quadratic displacement fields

AbstractThe classical lower and upper bound theorems allow the exact limit load for a perfectly plastic structure to be bracketed in a rigorous manner. When the bound theorems are implemented numerically in combination with the finite element method, the ability to obtain tight bracketing depends not only on the efficient solution of the arising optimization problem, but also on the effectiveness of the elements employed. Elements for (strict) upper bound analysis pose a particular difficulty since the flow rule is required to hold throughout each element, yet it can only be enforced at a finite number of points. For over 30 years, the standard choice for this type of analysis has been the constant strain element combined with discontinuities in the displacement field. Here we show that, provided certain conditions are observed, conventional linear strain triangles and tetrahedra can also be used to obtain strict upper bounds for a general convex yield function, even when the displacement field is discontinuous. A specific formulation for the Mohr–Coulomb criterion in plane strain is given in terms of second‐order cone programming, and example problems are solved using both continuous and discontinuous quadratic displacement fields. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Large‐scale deformation in the India‐Asia collision constrained by earthquakes and topography

AbstractAnalysis of earthquake focal mechanisms allows division of the India‐Asia collision into kinematic domains that strongly correlate with topography. These kinematic domains indicate strain partitioning dominated by oblique slip deformation. The Kunlun and south Tibetan fault systems mark discontinuities in the strain field and bound the high, flat topography of the plateau which deforms by transtension. The northern and southern margins of Tibet deform by transpression or contraction and are topographically steep. Correlations between seismicity and topography are due to Mohr–Coulomb wedge mechanics at the northern and southern plateau margins which produce naturally steep surface slopes, whereas the flat interior and eastern margin of the Tibetan Plateau is underlain by viscous crust which supports subdued topography further muted by Cenozoic basin fill. These data indicate that the long wavelength topography of the India‐Asia collision is controlled by seismically caused surface displacements which are linked to deep crustal deformation mechanics.

A consistent partly cracked XFEM element for cohesive crack growth

AbstractPresent extended finite element method (XFEM) elements for cohesive crack growth may often not be able to model equal stresses on both sides of the discontinuity when acting as a crack‐tip element. The authors have developed a new partly cracked XFEM element for cohesive crack growth with extra enrichments to the cracked elements. The extra enrichments are element side local and were developed by superposition of the standard nodal shape functions for the element and standard nodal shape functions for a sub‐triangle of the cracked element. With the extra enrichments, the crack‐tip element becomes capable of modelling variations in the discontinuous displacement field on both sides of the crack and hence also capable of modelling the case where equal stresses are present on each side of the crack. The enrichment was implemented for the 3‐node constant strain triangle (CST) and a standard algorithm was used to solve the non‐linear equations. The performance of the element is illustrated by modelling fracture mechanical benchmark tests. Investigations were carried out on the performance of the element for different crack lengths within one element. The results are compared with previously obtained XFEM results applying fully cracked XFEM elements, with computational results achieved using standard cohesive interface elements in a commercial code, and with experimental results. The suggested element performed well in the tests. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Multigrid based total variation image registration

We consider the image registration problem to find a reasonable displacement field, such that a transformed template image becomes similar to a so-called reference image. The minimization of the similarity measure (exemplarily based on the gray-value difference) yields a nonlinear ill-posed inverse problem. The necessary regularization is done by replacing the ill-conditioned Hessian by a multidimensional total variation $$(\mathcal{TV})$$ norm. This allows steep gradients and discontinuities in the displacement field in contrast to the common approach by elastic regularization which leads to globally smooth displacement fields. We propose and investigate a multigrid algorithm as inner iteration for $$\mathcal{TV}$$ registration. As we use Neumann boundary conditions which lead to singular systems, a special treatment before and during the FAS multigrid algorithm is required, e.g. the introduction and solution of an augmented system. We describe the necessary modifications for the multigrid algorithm and present convergence results as well as first registration experiments demonstrating the capabilities of the proposed approach.

A mesh-free simulation of cracking and failure using the cohesive segments method

An approach to the simulation of the cracking and failure of structures is presented that combines the mesh-free method with a cohesive segments model, in which cracking and failure are regarded as the appearance and combination of a series of cohesive segments. This method removes the need for explicit crack representation, and thus overcomes some of the limitations of conventional modeling methods. The entire displacement field is decomposed into a regular continuous field and various discontinuous fields that correspond to the cohesive segments. Using moving least-squares (MLS) shape functions as the partition of unity, the discontinuity field is approximated with extra degrees of freedom at the existing nodes. An iteratively solving scheme between the two fields is presented, and the chosen numerical examples show that the method is both flexible and efficient.

Two-dimensional multi-level strain estimation for discontinuous tissue

A large number of the strain estimation methods presented in the literature are based on the assumption of tissue continuity that establishes a continuous displacement field. However, in certain locations in the body such as the arteries in vivo scanning may produce displacement fields that are discontinuous between the two walls of the artery. Many of the displacement or strain estimators fail when the displacement fields are discontinuous. In this paper, we present a new 2D multi-level motion or displacement tracking method for accurate estimation of the strain in these situations. The final high-resolution displacement estimate is obtained using two processing steps. The first step involves an estimation of a coarse displacement estimate utilizing B-mode or envelope signals. To reduce computational time, the coarse displacement estimates are obtained starting from down-sampled B-mode pre- and post-compression image pairs using a pyramidal processing approach. The coarse displacement estimate obtained from the B-mode data is used to guide the final 2D cross-correlation computations on radio-frequency (RF) data. Results from finite element simulations and in vivo experimental data demonstrate the feasibility of this approach for imaging tissue with discontinuous displacement fields.

Extended parametric meshless Galerkin method

The parametric meshless Galerkin method (PMGM), which has recently emerged in the field of meshless methods, enhances the efficiency of meshless methods through the use of meshless shape functions which are mapped from a parametric space to the physical space. This property helps to save the running-time needed to calculate the meshless shape functions, which is a time consuming and complicated task. However, the meshless shape functions need to be regenerated in the vicinity of the boundaries as well as some other locations after mapping. In this paper, a new extension of the PMGM, called extended parametric meshless Galerkin method (X-PMGM) is presented. In this method, it is not necessary to regenerate the meshless shape functions near the boundaries, and instead, the approximation is enriched using discontinuous fields through a partition of unity method. Therefore the amount of shape functions regeneration is decreased which improves the efficiency of the method. Numerical examples show that very accurate results and computational time savings of about 80% are achieved comparing with the element-free Galerkin method results.

An extended arbitrary Lagrangian–Eulerian finite element modeling (X-ALE–FEM) in powder forming processes

Abstract In this paper, a new computational technique is presented based on the extended arbitrary Lagrangian–Eulerian finite element model (X-ALE–FEM) in plasticity forming of powder compaction. An arbitrary Lagrangian–Eulerian (ALE) technique is employed to capture the advantages of both Lagrangian and Eulerian methods and alleviate the drawbacks of the mesh distortion in Lagrangian formulation. In order to remove the limitation of the mesh conforming to the boundary conditions in this process, the extended finite element method (X-FEM) is implemented by incorporating discontinuous fields through a partition of unity method. The implementation of X-FEM technique in the framework of ALE model is finally presented in modeling of a shaped tablet component.

On singularities of continuity equation solutions

Generalized solutions to the continuity equation in one-dimensional and multidimensional cases are constructed in the case of a discontinuous velocity field.

Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics

In the present paper, the homogenized mechanical response of an interface in a microsystem interconnection is established on the basis of micropolar theory. The interface is treated as a finite RVE (representative volume element), across which macroscopic discontinuities occur as expressed in terms of the regularized discontinuous displacement and rotation fields. For the microstructure within the interfacial RVE, the micro–macro kinematical coupling is introduced as a second-order Taylor series expansion, along with a fluctuation term representing the microscopic displacement variation. In the second-order term of the expansion a restriction for the curvature is made, which motivates the adopted micropolar kinematics. Explicit expressions for the homogenized traction vector and the couple stress traction, associated with displacement and rotational jumps across the interface surface, are derived. A planar elastic interface is subjected to three basic deformation modes, i.e. the standard modes I, II and a non-conventional rotation mode, which are considered in the numerical examples representing a typical interconnect. A comparison to the results from the Taylor assumption is made, which shows that the Taylor assumption method produces an overstiffening of the interface.

Regularization of Discontinuous Vector Fields on $${\mathbb{R}^3}$$ via Singular Perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.