Unsymmetric Case Research Articles

Nonlinear bending of unsymmetric double-layer lattice truss core sandwich beams

This paper studied the nonlinear bending of unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). Five unsymmetric cases are considered by varying different materials and geometric thicknesses in two layers. To establish a theoretical model for the unsymmetric double-layer LTCSBs, Allen’s model and von Kármán nonlinear theory are employed. The present theoretical model is established by considering the midplane of the mid-sheet as the reference plane. Ritz method is applied to determine the deflection for the nonlinear bending, and verified by the finite element method (FEM). Finally, the effects of the loading direction, unsymmetric factors, and boundary conditions on the nonlinear bending behaviors of LTCSBs are studied. It is found that the nonlinear deflections of unsymmetric LTCSBs may exhibit softening-spring nonlinearity in one direction.

Alternate load path behaviour in structures with non-symmetrical design under column missing scenario

In the design and assessment of reinforced concrete (RC) structure against progressive collapse via single column removal scenario, also known as “alternate load path method”, development of secondary mechanisms, i.e. compressive arch action (CAA) and catenary action (CA) in double-span beams has been investigated and identified in many previous researches. The majority of the experimental tests are conducted on symmetrical double-span beams (identical span length and reinforcement detailing at both beam span). However, variations in details of beams adjacent to one another are not uncommon in RC building designs and constructions. Hence, a research study (as presented in this paper) with total of 5 RC frame tests are conducted to investigate the development of secondary mechanisms (CAA and CA) in a double-span beam with unsymmetrical 1) span length and 2) beam detailing. For each unsymmetrical case (span length and beam detailing), there will be 1 unsymmetrical double-span beam, and 2 symmetrical double-span beams (each representing one side of the unsymmetrical double-span beam) to serve as reference for quantifying the development of secondary mechanism in unsymmetrical double-span beams. The behaviour and development of secondary mechanisms are thoroughly analysed based on test observation, measured forces and displacements, etc.

Formula omitted] error estimates of unsymmetric RBF collocation for second order quasilinear elliptic equations

The paper proves the convergence of unsymmetric radial basis functions collocation for second order quasilinear elliptic equations. L2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. For the unsymmetric collocation case, we obtain the convergence result when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solution, the Lipschitz continuity of the Frechét derivative of quasilinear operator, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces.

Softening-spring nonlinearity in large amplitude vibration of unsymmetric double-layer lattice truss core sandwich beams

In recent years, many novel sandwich structures with multilayer or graded lattice truss cores that exhibited superior structural performance have been proposed. However, only limited works have studied the nonlinear behavior of sandwich structures with unsymmetric lattice truss cores. This paper aims to provide an analytical study on the nonlinear vibration of the unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). The double-layer LTCSB is designed to be unsymmetric and possesses varying material property and structural geometry in each layer. In this study, six unsymmetric cases of LTCSB classified as two categories according to the midplane locations are considered. Subsequently, an analytical model for the unsymmetric double-layer LTCSB is developed based on the Allen’s model and von Kármán nonlinear theory. The axial displacement of the midplane of LTCSB is considered in the analytical model, therefore the proposed model is more generalized compared with previous models for the symmetric double-layer LTCSB. The Ritz method with a direct iterative procedure is applied to solve the nonlinear governing equations and determine the nonlinear frequencies for the unsymmetric double-layer LTCSB. Finally, the effects of six unsymmetric cases of LTCSBs on the nonlinear frequency ratio versus amplitude curve under three different types of boundary conditions are discussed detailly. An interesting phenomenon of softening-spring nonlinearity is found for hinged–hinged and clamped–hinged sandwich beams with large bending–extension coupling.

Shutdown dose rate calculations with modified DEMO single sector model

A new approach for particle transport was used to calculate the Skyshine dose rates for the DEMO tokamak for the unsymmetrical case with one breeding blanket module removed for maintenance. Transport calculations for the large tokamaks, including DEMO and partly ITER, are usually performed in sector models postulating cyclic symmetry of geometry and source terms and applying reflecting boundary conditions on the sector edges. In the case of non-symmetric localized sources the sector models are not suitable.The calculation of shutdown dose rates, caused by one single tritium breeding blanket module moved from DEMO vacuum vessel into the upper maintenance hall, is an example of a localized radiation source and cannot be calculated in the standard way using a sector model. A new approach was followed for the calculations using the same sector model of DEMO, however with an according modification of the MCNP transport code. In this article the new approach is presented including its verification based on a simpler baseline model of DEMO. The results show that the obtained radiation dose distribution due to the one moved breeding blanket module differs by a factor of up to 2.8 from what would be obtained using a single sector model.

Semi-global stabilization of discrete-time linear systems with unsymmetrical saturation

This paper deals with the stabilization problem for discrete-time linear system with unsymmetric saturation. To handle the structural properties of the unsymmetric saturation nonlinearity, the description of the transformation carries from the symmetric case to the unsymmetric case is employed. By using the parametric Lyapunov equation-based low gain feedback design approach, linear matrix inequality formulations for the state feedback and output feedback control design are given. The effectiveness of the proposed method is confirmed by numerical examples.

A Recursive Methodology to Determine the Mechanical Response of Thin Laminated Plates in Bending

The paper’s objective is to present the development of a recursive methodology which is based on Adomian Decompostion Method in order to evaluate the mechanical response of thin laminated plates in linear bending. By the equivalent layer concept, the linear relation between the equivalent stresses and the strains, namely ABD matrix, is established. When viewed by the Adomian Decompostion Method perspective, it generates an interesting idea: each layer influence on the plate’s response can be recursively inserted into a base solution by a rearrangement of the plate’s properties. This base solution is previously obtained and it is, in this paper, an isotropic plate response for the same loading and boundary conditions. This approach can significantly increase optimization and delamination studies, given the simplicity on the layers modification, both in fiber orientation and constitutive properties, as these are considered on the recursive procedure. The pb-2 Rayleigh-Ritz Method is used to approximate the solution space and to generate analytic response surfaces. The methodology is applied to symmetrical and unsymmetrical stacking cases for different boundary conditions sets and loading types and the obtained responses are compared to those found on the literature. A study of case complements the methodology analysis: a simplified landing gear door is modeled considering a set of loading conditions as well as different stacking configurations. Good correspondence was found in all studied cases.

Moment redistribution in glass fiber reinforced polymer-reinforced concrete continuous beams subjected to unsymmetrical loading

This paper investigates the behavior of glass fiber reinforced polymer (GFRP)-reinforced concrete (RC) continuous beams subjected to unsymmetrical loading conditions. A total of six large-scale RC continuous beams were constructed and tested to failure. Three different loading cases were considered, (I) loading both spans equally, (II) loading both spans maintaining a load ratio of 1.5 and (III) loading one span only. The effects of reinforcing material type, assumed percentage of moment redistribution and serviceability requirements on the performance of the test beams were investigated. Also, a comparison between the experimental deflection and that predicted by available models, using effective moment of inertia approach, is presented. Test results revealed that, beams tested under unsymmetrical loading case II showed less crack widths, strains and deflections in the higher load span compared to those in the less load span or those symmetrically loaded. Moreover, the unsymmetrical loading conditions adversely affected the moment redistribution.

Accidental eccentricities, frame shear forces and ductility demands of buildings with uncertainties of stiffness and live load

Monte Carlo simulations of 3D structural models with 4, 6 and 15 stories, subjected to bidirectional seismic excitation, were generated to study the accidental torsional response of frame buildings. Variations of accidental eccentricities, story shear forces and story-drift ductility demands were studied. The probabilities of exceeding typical accidental-eccentricity recommendations of building codes were analyzed. Simulations assumed the following variables as random: (1) live-load magnitude, (2) live-load spatial distribution, and (3) flexural stiffness of columns and beams. The other structural variables, as well as the excitation, were considered as deterministic. Results show that, although the plan aspect ratio seems important for the selection of a design accidental eccentricity, this ratio does not have a significant influence on frame shear forces. Results indicate that maximum accidental eccentricities decrease as the building height decreases and that a normalized accidental eccentricity ea/b=0.05 seems acceptable for the tallest models studied, whereas a normalized eccentricity of ea/b=0.10 seems acceptable for shorter buildings. When an exceedance probability of 2% of frame shear forces is considered, results indicate that, for the unsymmetrical case, the frame shear force results roughly 20% larger than the computed force of the corresponding symmetrical case. Results also suggest that the design of regular frame buildings can be simplified by providing formulas to increase frame shear forces directly, instead of estimating and distributing building torsional moments among frames. The variations of both live loads and stiffness used in this study led to frame shear-force increments that varied between 10% and 40%, as compared with the nominally symmetrical case. The dispersion of the computed ductility demands caused by accidental eccentricity resulted small.

A generalized Block FSAI preconditioner for nonsymmetric linear systems

The efficient solution to nonsymmetric linear systems is still an open issue, especially on parallel computers. In this paper we generalize to the unsymmetric case the Block Factorized Sparse Approximate Inverse (Block FSAI) preconditioner which has already proved very effective on symmetric positive definite (SPD) problems. Block FSAI is a hybrid approach combining an “inner” preconditioner, with the aim of transforming the system matrix structure to block diagonal, with an “outer” one, a block diagonal incomplete or exact factorization intended to improve the conditioning of each block. The proposed algorithm is experimented with in a number of large size matrices showing both a good robustness and scalability.

Numerical Investigation for Fracture Saturation in Multilayer Sedimentary Rock in Unsymmetrical Case

Opening mode fractures in multilayer sedimentary rock often are periodically distributed with fracture spacing scaled to the thickness of the fractured layer. In this paper, based on Rock Failure Process Analysis Code RFPA2D, a three layer model with a central layer and with the different thickness top and bottom layer, progressive formation in multilayer sedimentary rock at fracture saturation in unsymmetrical case is simulated. We investigate the change of the critical fracture spacing to layer thickness ratio as a function of the thickness of the top layer where the bottom layers is much thicker (5 times) than the fractured layer called the unsymmetrical case, in this unsymmetrical case, fracture saturation is simulated. By numerical simulation of RFPA2D, the critical spacing to layer thickness ratio decreases and tend to the same constant value as the thickness of the top layer increases. Numerical simulation shown that for the unsymmetrical case, if the adjacent layers are thicker than 1.5 times the thickness of the fractured layer, the multilayer sedimentary rock can be treated approximately as a system with infinitely thick top and bottom layers at fracture saturation.That should be useful in the design of engineering systems and in the prediction of fracture spacing in hydrocarbon reservoirs and groundwater aquifers.

Differential qd Algorithm with Shifts for Rank-Structured Matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable, in particular, in the symmetric tridiagonal case where the differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy. In eigenvalue computations for rank-structured matrices, the QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, the QR algorithm destroys the rank structure and, hence, LR-type algorithms come in to play once again. In the current paper we adapt several variants of qd algorithms to quasi-separable matrices. Remarkably, one of them, when applied to Hessenberg matrices, becomes a direct generalization of the dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate and provides an alternative algorithm for finding roots of a polynomial represented in a basis of orthogonal polynomials. Results of preliminary numerical experiments are presented.

Qd-Type Methods for Quasiseparable Matrices

In the last few years many numerical techniques for computing eigenvalues of structured rank matrices have been proposed. Most of them are based on QR iterations since, in the symmetric case, the rank structure is preserved and high accuracy is guaranteed. In the unsymmetric case, however, the QR algorithm destroys the rank structure, which is instead preserved if LR iterations are used. We consider a wide class of quasiseparable matrices which can be represented in terms of the same parameters involved in their Neville factorization. This class, if assumptions are made to prevent possible breakdowns, is closed under LR steps. Moreover, we propose an implicit shifted LR method with a linear cost per step, which resembles the qd method for tridiagonal matrices. We show that for totally nonnegative quasiseparable matrices the algorithm is stable and breakdowns cannot occur if the Laguerre shift, or other shift strategy preserving nonnegativity, is used. Computational evidence shows that good accuracy is obtained also when applied to symmetric positive definite matrices.

On the numerical resolution of the discontinuous material bifurcation problem

AbstractThe work focuses on the numerical resolution of the discontinuous material bifurcation problem as a relevant ingredient in computational material failure mechanics. The problem consists of finding the conditions for the strain localization onset in terms of the so‐called bifurcation time, localization directions and localization modes. A numerical algorithm, based on the iterative resolution of a coupled eigenvalue problem in terms of the localization tensor, is proposed for such purpose. The algorithm is shown to be always convergent to the exact solution for the symmetric case (major and minor symmetries of the tangent constitutive operator). In the unsymmetric case (only minor symmetries), the solution is no longer exact, although it is shown that using the symmetric part of the localization tensor in the proposed algorithms provides enough accurate solutions for most of cases. Numerical examples illustrate the benefits of the proposed methodology in terms of accuracy and savings in the computational cost associated with the problem. Copyright © 2010 John Wiley & Sons, Ltd.

Investigation of the Dielectric-Loaded Folded Waveguide Traveling-Wave Tube Amplifier

The cold-test characteristics of the dielectric-loaded folded waveguide traveling-wave tube (FWTWT) amplifier are investigated theoretically and the Pierce small-signal theory is employed to confirm the results. For the purpose of analysis, the discussions are separated into symmetric case and unsymmetrical case according to the loading form of the dielectrics. The calculation results indicate that both of them, especially the unsymmetrical form, can significantly reduce the phase velocity and flatten the dispersion curve. However, loading dielectric in an unsymmetrical form will lead to a large decrease in the on-axis interaction impedance, thus it is recommended to use the symmetrical form. It is further found that when the thickness of the dielectric is small, i.e. h1/a ≤ 0.1, the interaction impedance is slightly affected by the variations on dielectric constant. The Pierce linear theory for a Q-band dielectric-loaded FWTWT amplifier shows that the bandwidth is increased from 15.08% to 27.03% and the operating voltage is decreased from 20.5 kv to 18.6 kv.

Thermodynamics of the Generalized Spin-One-Half Falicov-Kimball Model in Two Dimensions

The extrapolation of small-cluster exact-diagonalization calculations and the Monte Carlo method is used to study the spin-one-half Falicov–Kimball model extended by the spin-dependent Coulomb interaction (J) between the localized f and itinerant d electrons as well as the on-site Coulomb interaction (Uff ) between the localized f electrons. It is shown that in the symmetric case the ground-state phase diagram of the model has an extremely simple structure that consists of only two phases, and namely, the charge-density-wave phase and the spin-density-wave phase. The nonzero temperature studies showed that these phases persist also at finite temperatures. The same calculations that we performed for unsymmetric case showed that charge and spin ordering can be destroyed simultaneously or consecutively.

Algorithmic Aspects of Elimination Trees for Sparse Unsymmetric Matrices

The elimination tree of a symmetric matrix plays an important role in sparse elimination. We recently defined a generalization of this structure to the unsymmetric case that retains many of its properties. Here we present an algorithm for constructing the elimination tree of an unsymmetric matrix and show how it can be used to find a symmetric reordering of the matrix into a recursive, bordered block triangular form. We also present two symbolic factorization algorithms that use the elimination tree to determine the nonzero structures of the triangular factors of such matrices. Numerical experiments demonstrate that these algorithms are efficient and compare the new symbolic factorization schemes with existing ones.

High performance sparse solver for unsymmetrical linear equations with out-of-core strategies and its application on meshless methods

A new direct method for solving unsymmetrical sparse linear systems(USLS) arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin (MLPG) method need to solve large USLS. The proposed solution method for unsymmetrical case performs factorization processes symmetrically on the upper and lower triangular portion of matrix, which differs from previous work based on general unsymmetrical process, and attains higher performance. It is shown that the solution algorithm for USLS can be simply derived from the existing approaches for the symmetrical case. The new matrix factorization algorithm in our method can be implemented easily by modifying a standard JKI symmetrical matrix factorization code. Multi-blocked out-of-core strategies were also developed to expand the solution scale. The approach convincingly increases the speed of the solution process, which is demonstrated with the numerical tests.

Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

For a sparse symmetric matrix, there has been much attention given to algorithms for reducing the bandwidth. As far as we can see, little has been done for the unsymmetric matrix A, which has distinct lower and upper bandwidths l and u. When Gaussian elimination with row interchanges is applied, the lower bandwidth is unaltered, while the upper bandwidth becomes $l+u$. With column interchanges, the upper bandwidth is unaltered, while the lower bandwidth becomes $l+u$. We therefore seek to reduce $\min (l,u)+l+u$, which we call the total bandwidth. We compare applying the reverse Cuthill–McKee algorithm to $A+A^T$, to the row graph of A, and to the bipartite graph of A. We also propose an unsymmetric variant of the reverse Cuthill–McKee algorithm. In addition, we have adapted the node‐centroid and hill‐climbing ideas of Lim, Rodrigues, and Xiao to the unsymmetric case. We have found that using these to refine a Cuthill–McKee‐based ordering can give significant further bandwidth reductions. Numerical results for a range of practical problems are presented and comparisons made with the recent lexicographical method of Baumann, Fleischmann, and Mutzbauer.

Coherent beam-beam tune shift of unsymmetrical beam-beam interactions with large beam-beam parameter

Coherent beam-beam tune shift of unsymmetrical beam-beam interactions was studied experimentally and numerically in HERA, where the lepton beam has such a large beam-beam parameter (up to xi(y) = 0.272 ) that the single-particle motion is locally unstable at the origin (closed orbit). Unlike the symmetrical case of beam-beam interactions, the ratio of the coherent beam-beam tune shift and the beam-beam parameter in this unsymmetrical case of beam-beam interactions was found to decrease monotonically with an increase of the beam-beam parameter. The results of self-consistent beam-beam simulation, the linearized Vlasov equation, and the rigid-beam model were compared with the experimental measurement. It was found that the coherent beam-beam tune shifts measured in the experiment and calculated in the simulation agree remarkably well but they are much smaller than those calculated by the linearized Vlasov equation with the single-mode approximation or the rigid-beam model. The study indicated that the single-mode approximation in the linearization of the Vlasov equation is not valid in the case of unsymmetrical beam-beam interactions. The rigid-beam model is valid only with a small beam-beam parameter in the case of unsymmetrical beam-beam interactions.