J-integral for 3-D with center-cracked plate tests

A modification of the computer code ADINA[1] to calculate J-integrals for 2D isoparametric modelling was obtained from an EPRI-GE development project [2]. This coding was implemented and then revised to apply to a selected plane in 3D isoparametric finite element modeling. A previous dynamic fracture mechanics analysis of a center cracked plate [3] (using WECAN[4]) was modelled with ADINA[5] and used to test the special 2D and 3D J-integral features. Results included practical zeroes for closed crack values and volume independence in 3D (for a special static version of the test case), and excellent agreement between 2D and 3D for J-value dynamic histories. Good agreement was obtained between J-values here and J-values derived from stress intensity factors in the previous analysis [3]. This development in ADINA is a tool of considerable potential usefulness becase of many available features such as non-linear material properties.     

On application of differential geometry to computational mechanics

Developments in the fields of computational science—the finite element method—and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13–15] or modeling mechanical features of engineering problems under consideration [4–7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry.

Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. ³. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or C^r differentiable atlases which are compatible.

In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas.

The properties of geodesic curves have been used by Yuan et al. [13–15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four- and eight-node elements in ². The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in ³.

Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems.

Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in ² for which geodesic coordinates are used as local coordinates.     

Analysis of the hexcan of a fast breeder reactor with a through crack

Application of the finite element method to the problem involving a superficial crack in a hexcan of a fast breeder reactor is discussed.

The aim is to calculate the stress intensity factor of the hexcan with a through crack in the static and dynamic case, loaded with internal pressure. In the dynamic fracture the crack is considered both fixed and propagating. The structure is discretized with shell elements of 6 and 8 nodes available in the ADINA Code.     

Application of ADINA to nonlinear analysis of reinforced concrete shear walls with openings

This paper is concerned with application of ADINA to elasto-plastic analysis of the shear walls with openings. The authors analyzed the types of structures. One is the shear wall with many openings (the model of a secondary shield wall in nuclear power plant), on which scale model experiments were made. The other is the shear wall with openings in concrete rigid frame (the model of a shear wall in a building), on which parametric study was made.

In both cases, concrete is modeled using 8 nodes isoparametric 2 dimensional plane stress elements, reinforcing steels are modeled as truss elements. Concrete and elasto-plastic models are adopted for non-linear material model of concrete and reinforcing steel, respectively. The total numbers of nodes are 248–308, and that of 2D elements are 66–80.

Both analytical results are satisfactory from the view point of structural design. Close agreement to experimental results in the cracking load, crack extension, elasto-plastic stiffness and total strength was verified.     

Relation-algebraic semantics

The first half is a tutorial on orderings, lattices, Boolean algebras, operators on Boolean algebras, Tarski's fixed point theorem, and relation algebras.

In the second half, elements of a complete relation algebra are used as “meanings” for program statements. The use of relation algebras for this purpose was pioneered by de Bakker and de Roever in [10–12]. For a class of programming languages with program schemes, single μ-recursion, while-statements, if-then-else, sequential composition, and nondeterministic choice, a definition of “correct interpretation” is given which properly reflects the intuitive (or operational) meanings of the program constructs. A correct interpretation includes for each program statement an element serving as “input/output relation” and a domain element specifying that statement's “domain of nontermination”. The derivative of Hitchcock and Park [17] is defined and a relation-algebraic version of the extension by de Bakker [8, 9] of the Hitchcock-Park theorem is proved. The predicate transformers wp_s(-) and wlp_s(-) are defined and shown to obey all the standard laws in [15]. The “law of the excluded miracle” is shown to hold for an entire language if it holds for that language's basic statements (assignment statements and so on). Determinism is defined and characterized for all the program constructs. A relation-algebraic version of the invariance theorem for while-statements is given. An alternative definition of intepretation, called “demonic”, is obtained by using “demonic union” in place of ordinary union, and “demonic composition” in place of ordinary relational composition. Such interpretations are shown to arise naturally from a special class of correct interpretations, and to obey the laws of wp_s(-).     

Investigation of different isoparametric axisymmetric crack tip elements applied to a complete circumferential surface crack in a pipe

Standard isoparametric finite elements can be used as special crack tip elements in fracture mechanical computations by appropriately shifting the middle nodes in the neighbourhood of the crack tip. Such elements have already been applied to several plane and three-dimensional problems so that this method can be considered as commonly well accepted. In this paper the application of isoparametric axisymmetric elements as crack tip elements to a particular axisymmetric problem is studied. For that reason a complete circumferential crack at the inner surface of a pipe under axial tension is considered. The calculated stress intensity factors are compared with results from the literature. The general purpose finite element programs ASKA and ADINA have been used. In the first case triangular and quadrilateral elements were investigated, in the latter case quadrilateral and collapsed quadrilateral elements. In spite of the rather coarse grids good results for the stress intensity factor were found with the only exception of the collapsed quadrilateral elements.     

Simulation and automation of calculations of buildings (structures) on seismic effects

In many countries there are standards as one-dimensional, console systems for calculations of buildings, which stay in contradiction with the experience of destructive earthquakes. This scientific work offers an explanation for the transfer from one-dimensional to three-dimensional models of different complexity in standards. Discretely-continuous and discrete models of buildings have already been worked out as unified three-dimensional systems with floors deforming in their own plane.

We have considered models of seismic effects, which take into account the effect of the running seismic wave under the ground and the effect of non-uniformity of oscillation field along an extended building or a structure. We pointed out paradoxes in calculations while using three-dimensional models of buildings and “zero-dimensional” (normative) models of soil seismic effects on their grounds. In relation to this problem, we have executed the correction of the formula for determination of seismic forces.

Variational methods of making up the solving equations have been developed for calculation of oscillations of buildings as dynamic systems of large dimension. The structural analysis of buildings, of hydraulic structures, of bridges and of ship hulls has expediently proved to choose own vectors of rigidity matrixes separated out of a three-dimensional object of flat elements as possible displacements. It is the key to “rolling-up” extensive solving equations with thousands of the unknowns and “compressing” the three-dimensional object in one or two directions.

In order to simplify the calculations, we use so-called principle of partial symmetry, connected with mechanisms of deformation of cross- or longitudinal-sections of a three-dimensional construction. The principle can be considered as transformations leaving mathematical objects (tensors) invariant. In mathematical plan the symmetry causes break-up of solving equations into the independent blocks.

In an effort to adjust sequentially solutions, a hierarchical chain of mathematical simulation models of different levels was built, in which own vectors of flat elements are considered as hypotheses of deformation. We have developed program complex “PRIS” to automize calculations of buildings by three-dimensional models. The spectral methods of calculations are generalized for universal three-dimensional models of buildings to use standards of different countries in the developed program complex.     

Mixed mode fracture analysis of rectilinear anisotropic plates using singular boundary elements

This paper reports a multidomain boundary element formulation for direct calculation of stress intensity factors in rectilinear anisotropic plates subjected to arbitrary in-plane loading. The √r displacement and 1/√r traction behaviour near a crack tip is correctly represented in crack elements and transition elements. The use of these singular boundary elements is investigated for mode I and mixed mode crack problems.     

Spatially referenced methods of processing raster and vector data

The authors consider a general method of constructing addressing and arithmetic systems for two-dimensional image data using the hierarchy of ‘molecular’ tilings based on an original isohedral ‘atomic’ tiling. (Each molecular title at level k is formed from a constant number of tiles at level k−1; this is termed the ‘aperture’ property of the hierarchy.) In addition they present 11 objective criteria (which are of significance in cartographic image processing), by which these hierarchies and tilings may be described and compared.

Of the 11 topologically distinct types of isohedral tiling, three ([3⁶], [4⁴] and [6³]) are composed of regular polygons, and two of these ([3⁶] and[4⁴]) satisfy the condition that all tiles have the same ‘orientation’. In general, although each level in a hierarchy is topologically equivalent, the tiles may differ in shape at different levels and only [6³], [4⁴], [4.8²] and [4.6.12] are capable of giving rise to hierarchies in which the tiles at all levels are the same shape. The possible apertures of hierarchies obeying this condition are n² (for any n > 1)in the cases of [6³] and [4⁴]; n² or 2n² in the case of [4.8²]; and n² or 3n² in the case of [4.6.12].

In contrast the only tiling exhibiting the uniform ‘adjacency’ criterion is[3⁶]. However, hierarchies based on this atomic tiling generate molecular tiles with different shapes at every level. If these disadvantages are accepted, hierarchies based on first-level molecular tiles referred to as the 4-shape, 4′-shape, 7-shape and 9-shape are generated. Of these the 4-shape and the 9-shape appear to satisfy many of the cartographically desirable properties in addition to having an atomic tiling which exhibits uniform adjacency.

In recent years the generalized balanced ternary addressing system has been developed to exploit the image processing power of the 7-shape. The authors have generalized and extended this system as ‘tesseral addressing and arithmetic’, showing how it can be used to render a 4-shape into a spatially correct linear quadtree.     

A singular ES-FEM for plastic fracture mechanics

The stress and strain fields around the crack tip for power hardening material, which are singular as r approaches zero, are crucial to fracture and fatigue of structures. To simulate effectively the strain and stress around the crack tip, we develop a seven-node singular element which has a displacement field containing the HRR term and the second order term. The novel singular element is formulated based on the edge-based smoothed finite element method (ES-FEM). With one layer of these singular elements around the crack tip, the ES-FEM works very well for simulating plasticity around the crack tip based on the small strain formulation. Two examples are presented with detailed comparison with other methods. It is found that the results of the presented singular ES-FEM are closer to reference solution, which demonstrates the applicability and the effectiveness of our method for the plastic field around the crack tip.     

Fracture mechanics J-integral calculations in thermo-elasto-plasticity

A finite element post-processor has been developed to calculate an incremental plasticity-based J-integral for fracture mechanics evaluations. The post-processor accounts for elastic-plastic deformations and thermal strains. The ADINA finite element computer program, with minor modifications by Babcock and Wilcox, was used with the Ramberg-Osgood stress-strain law and provides through its “porthole” files the required results of stresses, strains, displacements, and elastic and plastic strain energies.

The numerical results of the post-processor indicate that the thermal J-integral, which consists of a line integral for the isothermal case and an additional area integral for the thermal effect, can be considered path-independent even in the presence of plastic and thermal strains.     

A finite element analysis of the finite width interface mixed mode bi-material fracture problem

One aspect of the terminal crack, mixed mode bi-material fracture mechanics problem is investigated using finite elements. The influence of a finite width bond line interface is considered for one representative material pair combination (E₂/E₁ = 0.10). The stress intensity factors for an inclined crack terminating at a variable thickness interface are established as a function of crack inclination. Since the order of the stress singularity is not the typical r^−1/2 associated with LEFM problems, variable power singular finite elements are used to model the terminating crack tip. Crack tip stress distributions and probable angles of crack extension are presented as a function of crack inclination and bond line thickness. Crack tip stress distributions assuming an interfacial debonding criterion are presented as a function of crack inclination and bond line thickness.     

Solution of elastic crack problems by superposition of finite elements and singular fields

A new finite element method is presented for the solution of plane elasticity problems which contain nonremovable stress singularities. Singular stress field are combined with finite element solutions by a superposition technique; an important feature of the method is that use of the singular fields may be restricted to any specified group of elements which include the singular point. It is shown that good estimates for stress intensity factors are obtained when the method is applied to crack problems.     

Construction of a hermite rational “Wachspress type” finite element

This paper concerns the construction of a quadrilateral finite element whose interpolation space admits of rational fractions for basis functions of “Wachspress type” [1, 2]. The construction of this finite element, which is in a way the “rational” equivalent of the ADINI finite element[3, 4], is founded on a method analogous to the one used for Serendip degree-two finite element construction in[2]. The study of interpolation error is dealt with in a paper by Apprato, Arcangeli and Gout in this journal “Rational interpolation of Wachspress error estimates”.     

Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.     

Three-dimensional J-integral calculations of part-through surface crack problems

In the field of nonlinear fracture mechanics, the J-integral is well known to play a significant role in predicting crack propagation and fracture strength. In this paper, we develop the three dimensional form of the J-integral using Eshelby's energy momentum tensor on any closed surface surrounding a crack front. Twenty-node isoparametric hexahedra elements in ADINA are used to calculate stress and strain distributions and Gaussian surface integration is used to evaluate J-integral values. Using the above method, we calculate the stress intensity factors for part-through surface cracks and verify the accuracy of these results.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

A simpler formula for the singular values of a certain Hankel operator

This paper deals with the problem of computing the singular values and vectors of a Hankel operator with symbol m^*W where m ε H^∞ is arbitrary inner and W ε H^∞ is rational. A simplified version of the formula given in [6] is obtained for computing the singular values of the Hankel operator. This result is applied to the (one-block) H^∞ optimal control of SISO stable infinite dimensional plants and rational weights. Using this new formula a simple expression is derived for the H^∞ optimal controller whose structure was observed in [9].     

Isoparametric finite element analysis for solid rocket motors

Several large scale finite element computerprograms have been written for the stress analysis of solid propellant rocket motors during the past several years. The displacement formulation is employed, and continuum and shell elements are coupled together along the idealized interface of the grain or insulation and the motor case. However, since the continuum elements exhibit a linear variation in displacements along each edge, and the shell elements have a cubic variation in displacements along their length, gaps can develop at the common interface of a coupled continuum and shell element. This incompatibility may lead to errors in stress predictions at a very important area in the solid rocket motor, the case/liner or case/ grain bondline.

In this present paper, the problem of grain/case finite element incompatibility is addressed within the framework of the conventional displacement formulation. Both the shell elements and the continuum elements are derived from the same quadratic isoparametric shape function, yielding finite element idealizations which are totally compatible along the interface of a continuum and shell element. A numerical example is presented which shows the differences between the present compatible formulation and the conventional incompatible formulations.     

Large-scale simulations of self-organizing neural networks on parallel computers: application to biological modelling

In this contribution we report about a study of a very versatile neural network algorithm known as “Self-organizing Feature Maps” and based on earlier work of Kohonen [1,2]. In its original version, the algorithm addresses a fundamental issue of brain organization, namely how topographically ordered maps of sensory information can be formed by learning.

This algorithm is investigated for a large number of neurons (up to 16 K) and for an input space of dimension d900. To meet the computational demands this algorithm was implemented on two parallel machines, on a self-built Transputer systolic ring and on a Connection Machine CM-2.

We will present below

1. (i) a simulation based on the feature map algorithm modelling part of the synaptic organization in the “hand-region” of the somatosensory cortex,

2. (ii) a study of the influence of the dimension of the input-space on the learning process,

3. (iii) a simulation of the extended algorithm, which explicitly includes lateral interactions, and

4. (iv) a comparison of the transputer-based “coarse-grained” implementation of the model, and the “fine-grained” implementation of the same system on the Connection Machine.