Reliability of the finite element method for calculating free edge stresses in composite laminates

The edge-stress problem for a [±45]_s graphite/epoxy laminate was examined in detail. A review of the literature on this problem showed that the interlaminar normal stress σ_z distributions along the interface between the +45° and -45° plies, obtained by various investigators, disagreed in magnitude and sign. In particular, a finite difference solution and a perturbation solution predicted a tensile σ_z, whereas the finite element methods predicted a compressive stress. Since a stress singularity exists at the intersection of the interface and the free edge, the differences in magnitude of the peak stress were expected, but not the difference in the sign.This paper investigates the reliability of the displacement-formulated finite element method in analyzing the edge-stress problem. Analyses of two well-known elasticity problems, one involving a stress discontinuity and one a singularity, showed that the finite element analysis yields accurate stress distributions everywhere except in two elements closest to the stress discontinuity or singularity. Stress distributions for a [±45]_s laminate showed the same behavior near the singularity as found in the well-known problems with exact solutions. The displacementformulated finite element method, therefore, appears to be a highly accurate technique for calculating interlaminar stresses in composite laminates. The disagreement among the numerical methods was attributed to the unsymmetric stress tensor at the singularity.     

Characterization of delamination effect on composite laminates using a new generalized layerwise approach

A dynamic analysis method has been developed to investigate and characterize the effect due to presence of discrete single and multiple embedded delaminations on the dynamic response of composite laminated structures with balanced/unbalanced and arbitrary stacking sequences in terms of number, placement, mode shapes and natural frequencies. A new generalized layerwise finite element model is developed to model the presence of multiple finite delamination in laminated composites. The new theory accurately predicts the interlaminar shear stresses while maintaining computational efficiency.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

A three-layered sandwich element with improved transverse shear stiffness and stresses based on FSDT

For analysing deformation and stresses of sandwich structures a displacement model is developed using the first-order shear deformation theory (FSDT) for each of the three layers, the core as well as the two face sheets. An enhancement of the Extended 2D Method by Rolfes and Rohwer [Improved transverse shear stresses in composite finite elements based on first order shear deformation theory. Int J Numer Methods Engrg 1997;40:51–60] is used to calculate improved transverse stiffness and stresses. Since the numerical effort is comparatively small and only C⁰-continuity is required for the shape functions, the theory is suitable for finite element applications. Within this paper a corresponding three-layered finite element is presented. The performance and the applicability of the proposed element are evaluated by considering numerical examples and by comparing with other two-dimensional models.     

On the structure of free finite state machines

The finite state machine U_{m, n}(M) freely generated by a set consisting of m states and n inputs subjects to the relations holding in the finite state machine M was considered by Birkhoff and Lipson in [1, 2]. In this paper, necessary and sufficient conditions for U_{m, n}(M) to consist of m disjoint copies of U_{1, n}(M) are established. The relationship between U_{1, n}(M) and the transition monoid of M, and a representation of U_{1, n}(M) as a transition monoid machine are described. The characterization of machines of type U_{1, n}(M) is in this way reduced to the characterization of finite monoids possessing a ‘universal presentation’. Some general results concerning finite semigroups and groups with a universal presentation, and precise characterizations of finite semilattices and Abelian groups admitting a universal presentation are described.     

A generalization of the Schützenberger product of finite monoids

For each n?1, an n-ary product ? on finite monoids is constructed. This product has the following property: Let Σ be a finite alphabet and Σ¹ the free monoid generated by Σ. For i = 1, …,n, let A_i be a recognizable subset of Σ¹, M(A_i) the syntactic monoid of A_n and M(A₁?A_n) the syntactic monoid of the concatenation product A₁?A_n. Then M(A₁?A_n)< ? (M(A₁),…,M(A_n)). The case n = 2 was studied by Schützenberger. As an application of the generalized product, I prove the theorem of Brzozowski and Knast that the dot-depth hierarchy of star-free sets is infinite.     

C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures

The aim of this work is to analyze the geometrically nonlinear mechanical behaviour of multilayered structures by a high order plate/shell finite element in order to predict displacements and stresses of such composite structures for design applications. Based on a conforming finite element method, a C¹ triangular six node finite element is developed using trigonometric functions for the transverse shear stresses. The geometric nonlinearity is based on von-Karmann assumptions and only five generalized displacements are used to ensure:

• •a cosine distribution for the transverse shear stresses with respect to the thickness co-ordinate, avoiding shear correction factors;
• •the continuity conditions between layers of the laminate for both displacements and transverse shear stresses;
• •the satisfaction of the boundary conditions at the top and bottom surfaces of the shells.

     

Data Rearrangement between Radix-k and Lee Distance Gray Codes in k-ary n-cubes

The k-ary n-cube is one of the popular topologies for interconnecting processors in multicomputers. This paper studies the difference in communication requirements between two Lee distance Gray codes when moving data from processors in normal radix k order to those in Gray code order in k -ary n-cube networks. Algorithms for k-ary to Gray code conversion, and vice versa, in k-ary n-cube networks are described under various channel constraints, i.e., one-port and all-port communication assumptions. The minimum length path routing algorithm for nonreflective Gray code requires roughly M(k/4) and (n−1) M(k/4) steps for data element transfers under all-port communication and one-port communication, respectively, for M elements per node. It is also shown that using a nonminimum length path routing algorithm, the number of steps for data element transfers can be halved. Lower bounds for the number of element transfers are derived, and the proposed algorithm using nonminimum length paths under one-port communication is shown to be near optimal.     

Strong convergence rate of estimators of change point and its application

Let {X_n,n?1} be an independent sequence with a mean shift. We consider the cumulative sum (CUSUM) estimator of a change point. It is shown that, when the rth moment of X_n is finite, for n?1 and r>1, strong convergence rate of the change point estimator is o(M(n)/n), for any M(n) satisfying that M(n)↑∞, which has improved the results in the literature. Furthermore, it is also shown that the preceding rate is still valid for some dependent or negative associate cases. We also propose an iterative algorithm to search for the location of a change point. A simulation study on a mean shift model with a stable distribution is provided, which demonstrates that the algorithm is efficient. In addition, a real data example is given for illustration.     

A high precision coupled bending–extension triangular finite element for laminated plates

It is well recognized that the estimation of interlaminar stresses and strain energy release rates is important in designing laminated composite panels. Generally coupled bending–extension finite elements are necessary to study laminates to include the effects of coupling and/or combined transverse and extensional loads. Such elements are normally formulated adapting the classical theory of bending and extension. While the classical laminated plate theory of bending has provision to obtain interlaminar stresses due to transverse loading, it is necessary to include certain higher order terms in the extensional theory in order to obtain the interlaminar stresses due to inplane loads. A high precision triangular element based on a theory which includes both the bending and extension with necessary higher order terms is presented in this paper. The performance of this element is validated with the aid of examples. Numerical results for displacements in symmetric and unsymmetric laminates under bending loads have been given. Numerical results for interlaminar stresses in symmetric and unsymmetric laminates have been given for the well-known benchmark problem of a coupon with free edges. Strain energy release rate components at the delamination tip in coupons with unsymmetric sublaminates have been given. The effects of delamination length and location on the components of the strain energy release rate have been studied. Results indicated that with the use of this element, the interlaminar stresses can be estimated reasonably accurately, over a major part of the laminate except in a small local region close to the free edge. Global–local analysis with three-dimensional elements in the local region, is suggested to obtain local stresses more accurately. Interlaminar stresses at the boundary of a hole in a perforated plate under extension have been obtained to illustrate the use of the present element in a global–local analysis strategy.     

Finite element analysis of the in-plane behaviour of annular disks

In-plane analysis of annular disks using the finite element method is presented. A semi-analytical, one-dimensional finite element model is developed using a Fourier series approach to account for the circumferential behaviour. Using displacement functions which are exact solutions of the two dimensional elasticity plane stress problem, the shape functions, stiffness matrices and mass matrices corresponding to the 0th, 1st and nth harmonics are derived. To show the utility of this new element, example probelms have been solved and compared with the exact solution. The present element can be readily coded into any general purpose finite element program.     

New approach in the determination of interlaminar shear stresses from the results of MSC/NASTRAN

A finite difference technique is applied to the strains and curvatures obtained from MSC/NASTRAN thin plate solutions to determine their derivatives. These quantities are then incorporated into classical thin plate theory to calculate interlaminar shear stresses. Several analyses are performed comparing the interlaminar shear stresses calculated by this method with those calculated theoretically and using MSC/NASTRAN beam theory approach. In all cases, the interlaminar shear stresses calculated by this method compare favorably with theoretical values. In addition, the interlaminar shear stresses are determined using MSC/NASTRAN beam theory approach and the method presented in this paper for a simply supported plate with a 30/-30/-30/30 layup subject to a uniform pressure. The technique presented in this paper can also be applied to other finite element programs.     

Mixed least-squares finite element model for the static analysis of laminated composite plates

This paper presents a mixed finite element model for the static analysis of laminated composite plates. The formulation is based on the least-squares variational principle, which is an alternative approach to the mixed weak form finite element models. The mixed least-squares finite element model considers the first-order shear deformation theory with generalized displacements and stress resultants as independent variables. Specifically, the mixed model is developed using equal-order C⁰ Lagrange interpolation functions of high p-levels along with full integration. This mixed least-squares-based discrete model yields a symmetric and positive-definite system of algebraic equations. The predictive capability of the proposed model is demonstrated by numerical examples of the static analysis of four laminated composite plates, with different boundary conditions and various side-to-thickness ratios. Particularly, the mixed least-squares model with high-order interpolation functions is shown to be insensitive to shear-locking.     

Asymmetric vibration and stability of circular plates

The asymmetric vibration and stability of circular and annular plates using the finite element method is discussed. The plate bending model consists of one-dimensional circular and annular ring segments using a Fourier series approach to model the problem asymmetries. Using displacement functions which are the exact solutions of the static plate bending equation, the stiffness coefficients corresponding to the 1st and nth harmonics are used in closed form. By assuming that the static displacement function closely represents the vibration and stability modes, the mass and stability coefficient matrices for an annular and circular element are also constructed for the 1st and nth harmonics. Several numerical examples are presented to demonstrate the efficiency and accuracy of the finite element model with that of classical methods.     

Characteristic stabilized finite element method for the transient Navier–Stokes equations

Based on the lowest equal-order conforming finite element subspace (X_h, M_h) (i.e. P₁–P₁ or Q₁–Q₁ elements), a characteristic stabilized finite element method for transient Navier–Stokes problem is proposed. The proposed method has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, and averting the difficulties caused by trilinear terms. Existence,uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method. Finally, a serious of numerical experiments are given to show that this method is highly efficient for transient Navier–Stokes problem.     

A finite element approach to global-local modeling in composite laminate analysis

A finite element model is described to study interlaminar stresses within polymer composite laminated materials. This model is based upon a global-local model proposed by Pagano and Soni in 1983. The development of solution procedures includes an out-of-core memory solving technique. The numerical results generated for simple plate problems with and without holes in the center under uniaxial loading are reported. Comments regarding the finite-element mesh-size, numerical stability, problem size and sensitivity of results to substructuring of the laminate into global and local regions have also been discussed.     

Application of the boundary-integral equation method to three dimensional stress analysis

The boundary-integral equation method is based on a mathematical formulation which reduces the dimensionality of a problem by relating surface tractions to surface displacements. Discretization of the surface allows a direct and standard algebraic solution for the unknown surface data. The stresses at any point are then found by direct quadrature from the entirety of surface data. Because of the reduction of the dimension of the problem, the size of the algebraic problem is considerably smaller than for finite element models. Also, since only the surface is discretized the analyst is able to achieve considerably greater resolution of interior stresses than by finite element models.The paper reports on two direct comparisons of the boundary-integral equation and finite element methods: two dimensional crack problems and a bulky three dimensional problem representative of mine structures. Discussion of the results will include accuracy, storage requirements and computer time comparisons. The paper also discusses the utility of the boundary-integral method to model three dimensional elastic bodies with through and part-through cracks.     

Mesh Optimization Using Global Error with Application to Geometry Simplification

This paper presents a linear running time optimization algorithm for meshes with subdivision connectivity, e.g., subdivision surfaces. The algorithm optimizes a model using a metric defined by the user. Two functionals are used to build the metric: a rate functional and a distortion (i.e. error) functional. The distortion functional defines the error function to minimize, whereas the rate functional defines the minimization constraint. The algorithm computes approximations within this metric using jointly global error and an optimal vertex selection technique inspired from optimal tree pruning algorithms used in compression. We present an update mechanism, that we name merging domain intersections (MDIs), allowing the control of global error through the optimization process at low cost. Our method has application in progressive model decomposition, compression, rendering, and finite element methods. We apply our method to geometry simplification and present an algorithm to compute a decomposition of a model into a multiresolution hierarchy in O(n log n) time using global error, where n is the number of vertices in the full-resolution model. We show that a direct approach, i.e. not using MDIs, recomputing global error has at least cost O(n²). We analyze the optimality of the algorithm and give several for its properties. We present results for semi-regular meshes obtained from approximation of subdivision surfaces whose connectivity is obtain from (triangulated) quadrilateral quadrisection (e.g. 4-8 or Catmull-Clark subdivision).     

The semiring of matrices over a finite chain

Let L_m denote the chain {0,1,2,…,m-1} with the usual ordering and M_n(L_m) the matrix semiring of all n×n matrices with elements in L_m. We firstly introduce some order-preserving semiring homomorphisms from M_n(L_m) to M(L_k). By using these homomorphisms, we show that a matrix over the finite chain L_m can be decomposed into the sum of some matrices over the finite chain L_k, where k<m. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1-25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring M_n(L_m) to the direct product of the h copies of the semiring M_n(L_k) and discuss Green’s relations on the multiplicative semigroup of the semiring M_n(L_m). We think that some results obtained in this paper is useful for the study of fuzzy matrices.