B. Daya Reddy

The finite element method

In practical situations the determination of suitable basis functions for use in the Galerkin method can be extremely difficult, especially in cases for which the domain Ω does not have a simple shape. The finite element method overcomes this difficulty by providing a systematic means for generating basis functions on domains of fairly arbitrary shape. What makes the method especially attractive is the fact that these basis functions are piecewise polynomials that are nonzero only on a relatively small part of Ω, so that computations may be carried out in a modular fashion, which is well suited to computer-based approaches. As we show a little later, the family of spaces V h (h ∈ (0, 1)) defined by the finite element procedure possesses the property that V h approaches V as h approaches zero, in an appropriate sense. This is, of course, an indispensable property for convergence of the Galerkin method.

On the finite element method for mixed variational inequalities arising in elastoplasticity

We analyze the finite-element method for a class of mixed variational inequalities of the second kind, which arises in elastoplastic problems. An abstract variational inequality, of which the elast...

A computational study of knitted Nitinol meshes for their prospective use as external vein reinforcement

External reinforcement has been suggested for autologous vein grafts to address the mismatch of mechanical properties and fluid dynamics of graft and host vessel, a main factor for graft failure. A finite-element tool was developed to investigate the mechanical behaviour, in particular radial compliance, of knitted Nitinol meshes (internal diameter: 3.34 mm) with two different knit designs (even versus uneven circumferential loops) and three different wire thicknesses (0.05, 0.0635 and 0.075 mm) under physiological conditions. The Nitinol material parameters were obtained from experimental testing. The compliance predicted for the 80-120 mmHg physiological blood pressure range was 2.5, 0.9 and 0.6%/100 mmHg for the even loop design and 1.2, 0.5 and 0.5%/100 mmHg for the uneven loop design, for wire thicknesses of 0.05, 0.0635 and 0.075 mm. The highest stress, at 120 mmHg, was found in the even loop mesh with the thinnest wire to be 268 MPa, remaining 44.5% below the stress initiating stress-induced phase transformation. The maximum stress decreased to 132 and 91 MPa with increasing wire thickness of the same loop design. The uneven loop design exhibited maximum stress levels of 65.3%, 63.6% and 87.9% of the even loop values at 0.05, 0.0635 and 0.075 mm wire thickness. The maximum strain of 0.7%, at 120 mmHg, remained un-critical considering a typical high-cycle recoverable strain of 2%. It was demonstrated that the numerical approach developed was feasible of effectively evaluating design variations of knitted Nitinol meshes towards vein graft behaviour equivalent to arterial mechanics.

Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity

Summary. This work considers semi- and fully discrete approximations to the primal problem in elastoplasticity. The unknowns are displacement and internal variables, and the problem takes the form of an evolution variational inequality. Strong convergence of time-discrete, as well as spatially and fully discrete approximations, is established without making any assumptions of regularity over and above those established in the proof of well-posedness of this problem.

Qualitative and Numerical Analysis of Quasi-Static Problems in Elastoplasticity

The quasi-static problem of elastoplasticity with combined kinematic-isotropic hardening is formulated as a time-dependent variational inequality (VI) of the mixed kind; that is, it is an inequality involving a nondifferentiable functional and is imposed on a subset of a space. This VI differs from the standard parabolic VI in that time derivatives of the unknown variable occur in all of its terms. The problem is shown to possess a unique solution. We consider two types of approximations to the VI corresponding to the quasi-static problem of elastoplasticity: semidiscrete approximations, in which only the spatial domain is discretized, by finite elements; and fully discrete approximations, in which the spatial domain is again discretized by finite elements, and the temporal domain is discretized and the time-derivative appearing in the VI is approximated in an appropriate way. Estimates of the errors inherent in the above two types of approximations, in suitable Sobolev norms, are obtained for the quasi-static problem of elastoplasticity; in particular, these estimates express rates of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, of the time step size k. A major difficulty in solving the problems is caused by the presence of the nondifferentiable terms. We consider some regularization techniques for overcoming the difficulty. Besides the usual convergence estimates, we also provide a posteriori error estimates which enable us to estimate the error by using only the solution of a regularized problem.

Well-posedness of the problem of fiber suspension flows

The well-posedness of the equations governing the flow of fiber suspensions is studied. The fluid is assumed to be Newtonian and incompressible, and the presence of fibers is accounted for through the use of second- and fourth-order orientation tensors, which model the effects of the orientation of fibers in an averaged sense. The fourth-order orientation tensor is expressed in terms of the second-order tensor through various closure relations. It is shown that the linear closure relation leads to anomalous behavior, in that the rest state of the fluid is unstable, in the sense of Liapounov, for certain ranges of the fiber particle number. No such anomalies arise in the case of quadratic and hybrid closure relations. For the quadratic closure relation, it is shown that a unique solution exists locally in time for small data.

Anatomical Plasticity in the Snout of Lystrosaurus

The skull of Lystrosaurus, characterized by an elongated snout and a scarf premaxilla-nasal suture, differs from the generalized Permian dicynodont form. The sutural relationships of the bones of the Lystrosaurus snout are further investigated here using several anatomical lines of evidence: gross osteology, histological and serial sections, and micro-computed tomography scans. Novel evidence was found for supernumerary bone(s) in the dorsal region of the snout in a few specimens of Lystrosaurus. The developmental and functional implications of this anatomical plasticity are discussed. It is hypothesized that the supernumerary bones may have formed from separate ossification centers of the frontal bone.

Convergence analysis of discrete approximations of problems in hardening plasticity

The initial boundary value problem of quasistatic elastoplasticity is considered here as a variational inequality and equation in the displacement and stress. A variational inequality for the stress only may be obtained by eliminating the displacement. Semidiscrete approximations of the stress problem and fully discrete finite element approximations of the full problem are considered under assumptions of minimum regularity of the solution. It is shown that the resulting families of approximations converge to the solution of the original problem.

A mathematical method for constraint-based cluster analysis towards optimized constrictive diameter smoothing of saphenous vein grafts

This study was concerned with the cluster analysis of saphenous vein graft data to determine a minimum number of diameters, and their values, for the constrictive smoothing of diameter irregularities of a cohort of veins. Mathematical algorithms were developed for data selection, transformation and clustering. Constrictive diameter values were identified with interactive pattern evaluation and subsequently facilitated in decision-tree algorithms for the data clustering. The novel method proved feasible for the analysis of data of 118 veins grafts, identifying the minimum of two diameter classes. The results were compared to outcome of a statistical recursive partitioning analysis of the data set. The method can easily be implemented in computer-based intelligent systems for the analysis of larger data sets using the diameter classes identified as initial cluster structure.

The four-noded quadrilateral with a 2 × 1 integration rule: Application to plates and other problems

Abstract In the finite element analysis of plane problems, the Gauss integration scheme employed in conjunction with the standard four-noded quadrilateral is conventionally that which uses 4 (= 2 × 2) points. Underintegration in this case amounts to the use of a single integration point. A disadvantage of the use of the standard 2 × 2 scheme is the lack of accuracy for coarse meshes, as well as locking in the incompressible limit. The use of one-point integration, on the other hand, has the disadvantage that the resulting stiffness matrix is rank-deficient, and has to be stabilized. This work explores an intermediate possibility, viz. a two-point integration scheme, in the context of plane problems. This element has been introduced in an earlier work, and has been shown to give remarkably good results. That investigation is carried further here, with the procedure being applied to problems of axisymmetry, to plate problems, and to nonlinear problems. For plate problems in particular, it is shown that a minor modification to the integration rule for the Bathe-Dvorkin element leads to a marked improvement in results.