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A field-enriched finite element method is proposed in this paper to simulate the failure process of rocks. According to the different types of defects, the proposed method can select the appropriate failure criteria for voids and cracks. Field variables are used to characterize the crack location and to realize the influence of cracks on the physical field of the numerical model. The main features and the capability of the field-enriched finite element method to efficiently simulate the failure process of rocks with different defects are illustrated with some numerical examples. The results show that the field-enriched finite element method proposed in this paper can obtain the crack propagation path which is consistent with other numerical methods and experimental observations. Moreover, the numerical method proposed in this paper has the advantages of simple implementation process and higher computational efficiency and accuracy in some cases. Therefore, it is a promising numerical simulation method.

In this paper, the authors carried out the optimization of an isogrid structure considering six different responses: Tsai-Wu failure index under compression and torsion efforts; instability coefficient under compression and torsion efforts; mass and natural frequency. The response surface methodology was used to find the functions that describe the model from the data found by numerical analysis; and a new optimization algorithm called SunFlower Optimization (SFO) was used to find the best possible configurations. This paper shows something unprecedented in the literature to date when evaluating the compression, torsion, and modal performance of an isogrid tube. The simultaneous evaluation of several responses brings something more real to the scientific world, the results obtained here suggest that the optimal configuration for one response can be very bad for others and the use of a multi-objective optimization solved this problem.

The paper describes various approaches for the mathematical modelling of Adaptive Inflatable Structures (AIS) along with the corresponding numerical methods. The introductory part presents a general idea of adaptive impact absorption (AIA) and the concept of inflatable structures equipped with controllable valves serving for internal pressure control. Application of AIS for adaptive absorption of the impact loading is briefly explained. The paper focuses on diverse methods of modelling of inflatable structures, which are based on interaction between solid walls and fluid enclosed inside. Modelling of the solid walls is based on rigid body dynamics or initial-boundary value problem of solid mechanics. In turn, modelling of the fluid utilizes either classical equilibrium thermodynamics or Navier-Stokes equations. Consequently, four possible combinations of the above approaches are distinguished, precisely analyzed and applied for the modelling of different types of inflatable structures. Each model takes into account controllable valves, which requires introducing additional coupling between parameters defining the valves and selected results of the analysis. Corresponding numerical methods include classical methods of solving ordinary differential equations, finite volume method (FVM) applied for problems with mobile boundaries, finite element method (FEM) applied for problems involving additional ODEs and, finally, FEM coupled with FVM. Proposed numerical methods and software tools are utilized for the simulation of adaptive pneumatic cylinders, adaptive pneumatic fenders and membrane valves.

A newly developed X-SBFEM based on the LSM is proposed in this paper for fracture mechanics.The level set method algorithm further develops the X-SBFEM and is used to solve the crack propagation problem.Large structure engineering practice based on XSBFEM.The numerical implementation of XSBFEM. The extended scaled boundary finite element method (X-SBFEM) based on the level set method (LSM) is proposed in this paper to combine the advantages of the scaled boundary finite element method (SBFEM) and the extended finite element method (XFEM). The level set method (LSM) algorithm is applied to further develop the X-SBFEM, especially for the crack propagation problem. The Heaviside enrichment function is used to represent a jump across a discontinuity surface in a split element, and the non-smooth behavior around the crack tip is described using the semi-analytical SBFEM. The stiffness of the region containing the crack tip is computed directly, and the generalized stress intensity factors of many types of singularities are obtained directly from their definitions using consistent formulas. In the numerical simulations, a square plate with an edge crack under tension, a three-point bending beam, a four-point shear beam and a dam (the Koyna dam) with a single propagating crack are modeled. The results show that the proposed X-SBFEM is capable of calculating the stress intensity factors of cracks and predicting crack trajectories and load-displacement relations accurately. An analysis of the sensitivity of the parameters is employed to demonstrate that various mesh densities and crack propagation step lengths led to consistent results.

Minimum weight formulations with stress constraints have been recently proposed as an alternative to the traditional maximum stiffness statements for the topology optimization of structures. These minimum weight approaches offer some advantages and avoid unwanted phenomena associated to maximum stiffness approaches, e.g. mesh dependency, checkerboard layouts. In addition, minimum weight formulations analyze more usual statements from a practical point of view in engineering since they reduce the cost of the structure and impose stress constraints. The minimum weight formulation with stress constraints proposed in this paper guarantees the feasibility of the optimal solutions obtained while the cost is minimized. However, these formulations also require bigger computing effort than the traditional maximum stiffness statements since the number of highly non-linear stress constraints is drastically increased while the number of design variables is analogous. Thus, it is necessary to introduce numerical methods and computational techniques that allow to reduce the resources required. In this paper, we propose the use of parallelization techniques in order to reduce the computing time required to solve the topology optimization problem with stress constraints proposed.

This paper presents the outcome of a large parametric numerical analysis of solid and hollow reinforced concrete piers taken from actually constructed bridges, based on a consistent three-dimensional nonlinear finite element methodology that was presented in a companion paper. Various transverse reinforcement arrangements and spacings were examined, as well as the effect of high-strength concrete on confinement effectiveness. The interpretation of numerical results mainly focuses on identifying the most convenient confinement configurations in terms of enhanced strength and ductility, as well as ease of construction and cost effectiveness. Furthermore, issues regarding confinement arrangements (often used in practice) that result in reduced section ductility are investigated and possible remedies are suggested. Finally, the broad applicability of the proposed methodology is established by application to a particularly complex (in terms of geometry and reinforcement detailing) hollow pylon section.

In this paper, a non-linear finite element framework has been implemented to simulate the sequential build-up of a flip-chip package. A generalized deformation model with element removal and addition is used to activate and deactivate the underfill material layer to simulate flip-chip package fabrication. Using process models, one can determine the warpage stresses at any intermediate stage in the process. In addition, topological change is also considered in order to model the sequential steps during the flip-chip assembly. Geometric and material nonlinearity which includes the creep behavior of underfill and solder balls, and temperature-dependent material properties are considered. Different stress-free temperatures for different elements in the same model are used to simulate practical manufacturing process-induced thermal residual stress field in the chip assembly. This approach (the processing model established in this paper) is in contrast to the non-processing model employed by many researchers, which is shown to yield overly conservative and sometimes erroneous results, leading to non-optimal design solutions. From the finite element analysis, it is found that the strains and deflections obtained from the non-processing model are generally smaller than those obtained from the processing model due to the negligence of the bonding process-induced residual strains and warpage. Furthermore, the fatigue life for the outmost solder ball predicted by the processing model is much shorter than that predicted by the non processing model based on the Coffin–Manson equation. On the other hand, in order to prove the soundness of the framework established in this paper, the test results obtained by using the laser moireinterferometry technique are compared with the results achieved from the proposed numerical analysis vehicle. It is shown that the deformation values of the flip-chip package predicted from the finite element analysis are in a good agreement with those obtained from the test.

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.

With the rapid development of advanced composite materials and the wide application of such materials in engineering, it is desirable to model problems involving these materials by computer and to use powerful numerical methods like the finite element method (FEM) and the boundary element method (BEM) for analysis. In this paper, a BEM is developed to analyze two-dimensional micromechanical behavior of fiber-reinforced composites based on models for both perfectly-bonded and imperfectly-bonded materials in a unit cell. For composites with perfect bond between matrix and fibers, it is shown that our predictions coincide satisfactorily with comparable quantities obtained in physical experiments and by FEM analysis. For imperfectly-bonded composites, it is found that variation of the interphase parameters (thickness, stiffness) causes pronounced changes in the overall effective moduli and also in the state of stress in the composites. Also in this paper, the idea of a library of Green's functions for fiber-reinforced composites is discussed. With such a library, users could quickly generate data useful in design with very little knowledge of methods for computational modeling in general or of the BEM in particular.

In a companion paper, the relationship between crack propagation stability and the second coefficient of the Williams stress function has been discussed and the program ANGCRK was validated. In this paper, ANGCRK has been used in the evaluation of crack propagation stability in edge-cracked plates under biaxial loading. For this purpose, the effects of the applied loads on different points is studied. Then the second coefficient of the Williams stress field of the problems at hand is calculated, identifying those cases in which crack propagation instability is evaluated. The numerical results are compared with the experimental results and good agreement is obtained. Also, the effect of the specimen geometry and loading conditions is analyzed.

The fourth order governing differential equations for thin-walled beams with asymmetric cross-sections under coupled bending and torsion are derived in this paper. The stiffness matrix and the consistent mass matrix of a thin-walled beam element with an asymmetric cross-section is established by using the homogeneous solutions to the fourth order governing differential equations as interpolation functions between the degrees of freedom at each node of the element. The shear effect and the coupling between bending and torsion are fully taken into account in the finite element model developed in this paper. This model can be used in static and dynamic analysis of the thin-walled beams with asymmetric cross-sections.

In computing eigenvalues for a large finite element system it has been observed that the eigenvalue extractors produce eigenvectors that are in some sense more accurate than their corresponding eigenvalues. From this observation the paper uses a patch type technique based on the eigenvector for one mesh quality to provide an eigenvalue error indicator. Tests show this indicator to be both accurate and reliable. This technique was first observed by the authors for an error estimation for the buckling and natural frequency of beams and for two-dimensional in plane structures. This paper produces an error indicator for the more complex problem of out-of-plane plate buckling analysis.

A previous paper [1] proposed an elastic tripping model using a Rayleigh-Ritz approach with 4 d.f. This paper extends the elastic model into the inelastic range, using deformation theory and an iterative and incremental formulation. Nonetheless, because of the efficiency of the Rayleigh-Ritz formulation, the solution is very rapid, even for ordinary PCs. The method can be used for analyzing the flexural-torsional and lateral-torsional buckling (“tripping”) behavior of flanged stiffeners subjected to axial compression, end moment, uniform lateral pressure and any combination of these. The effects of cross-sectional distortion, postbuckling behavior of the plate (incorporated by considering the plate effective width) and plasticity are included. Results obtained using the method are shown to be in good agreement with experimental results, and to be more accurate than other methods.

This paper focuses on studying the load distribution of multifastener composite joints under the applied shearing load or off-axis tensile load. In this paper, the models of multifastener joints for the finite element analysis are discussed. An idealization model under the shearing load is set up and the difference between two models under the shearing load and the tensile load are compared. The presented method in the paper can be used to compute bolt loadings, which include their magnitudes and directions. The radial forces on contact boundary can be solved by the displacement coordination equation. In illustrative examples the joints with four fasteners, six fasteners and eight fasteners in two rows under the shearing load are computed. The computation results are coincident with experiment data. In addition, this method is used to compute the load distribution of a joint with three fasteners in a row under off-axis tensile loading. The computation results show that the load distribution has many characteristics under off-axis loading compared with that under on-axis tensile load. The method presented by this paper can be used to predict in engineering.

In order to solve even simple plate problems, one may need to set up quite an amount of computer codes for the implementation of general numerical methods such as finite element analysis. Using the semi-analytical approach introduced in this paper, the amount of time-consuming work may be reduced significantly. It would be even more convenient if the solutions were given in the forms of algebraic expressions instead of numerical values. A primary concern in the design of plates would be the deflections of plates due to loadings at various points inside the plates. The deflections can be obtained from an influence function known as the Green's function of the plates. However, except a few known simple cases, it is difficult to obtain Green's function of the plate for various shapes and boundary conditions analytically. To circumvent such difficulties in finding this function, a numerical Green's function is derived in this paper for the first time. Once the numerical Green's function is obtained the solution, a lateral deflection in this paper, may be calculated by carrying out simple integration. Also, since the numerical Green's function obtained in this paper is given in terms of algebraic polynomials, it is suitable for differentiation or integration without introducing numerical errors using symbolic computation. Utilizing the numerical Green's functions of plates, a simple and accurate yet flexible approach to the subject is addressed.

This paper deals with the determination of stresses and strains in cross-sections that satisfy Navier's assumption. An original semi-analytical method to compute the surface integrals involved is presented. The distinguishing characteristic of this method, that leads to an easy, recurrent algebraic formula, is independence of the shape, both of the section and of the constitutive law adopted. Furthermore, the work proposes the problem related to sections cast in two distinct phases and shows a way this problem can be solved adopting the same subroutines developed to investigate sections cast in one single step. The discussion of some issues that influence the selection of the iterative method ends the paper.

The improved body force method (BFM), which can effectively analyze the interference effect of arbitrary multiple holes and cracks subjected to complex loadings, is proposed in this paper, based on the basic principles of the BFM developed by Nisitani. Three improvements are made with respect to the original BFM. First, through introducing the local coordinate system and modifying the definition of body force densities on a hole element, solutions for all stress influence coefficients induced by body force densities distributed over the hole element are derived in a closed form. Second, a local coordinate system is introduced at the center of each crack, and closed form solutions for some of the stress influence coefficients caused by body force doublet densities on a crack element are found. Third, based on the physical view of the stress influence coefficient, a novel and simple indirect method for solving the integral singularity of the crack problem is put forward and the results obtained can also be expressed in a closed form. As a result, not only the computational efficiency and precision of the BFM are improved, but also its engineering application is made more convenient and effective. Finally, as a comparison of the improved BFM and other methods, the paper presents a case analysis of several typical problems. The results show that the improved BFM proves itself to be excellent in higher precision and faster convergence compared to other methods when the interference effects of multiple holes and cracks are analyzed.

Recently nonlinear elements such as structures with different tension and compression moduli, supports with motion-limiting stops, and mechanical systems having clearances or snubbers, have been widely used to obtain better performance in vibration controls. When a spring is stretched or compressed beyond the linear range, the restoring force increases at a greater rate than the deformation for a “hardening spring” and at a smaller rate for a “softening spring”. Hence, such a simple mass-spring oscillator is to be regarded as a nonlinear system. The purpose of this paper is to introduce the application of the Taylor differential transformation method to the viscous damped vibration of a hardening and softening spring system. The method introduced in this paper offers advantages over most other methods in its generality, efficiency, accuracy and ease-of-use for calculating the dynamic response of the nonlinear system. The computation of the Taylor differential transformation is complex and difficult probably, but, in the practical application, its advantages are very evident, especially for nonlinear systems and the time-dependant parameter systems. The computationally accurate and the quickly converging calculation by a computer with the Taylor differential transformation method are very salient.

The paper presents a hybrid-Trefftz element approach for the numerical solution of transient linear heat conduction problems. In the proposed method, the transient heat conduction equation is first discretized with respect to time and then the resulting set of elliptic equations is solved by the corresponding time independent hybrid Trefftz element approach. At the end of the paper, the proposed method is assessed through numerical examples.

In a companion paper we presented a neural dynamics model for optimization of structures by integrating penalty function method, Lyapunov stability theorem, Kuhn-Tucker condition, and the neural dynamics concept. In this paper, we apply the model to optimum plastic design of low-rise steel frames. The objective and constraint functions are scaled to improve the efficiency and numerical conditioning of the algorithm. As demonstrated in the convergence histories for the four examples presented, the neural dynamics model yields stable results no matter how the starting point is selected. Since the neural dynamics model lends itself to concurrent processing effectively, development of a concurrent neural dynamics model for optimization of large structures appears a very promising approach, which is currently under investigation by the authors.

This paper presents a simplified model of buckling mechanism which occurs in slender steel column elements under compressive axial loading. This model has been used to predict buckling capacities of certain structures, and comparisons have been made with test results. The method presented is not as accurate as the current finite element methods which use complete large displacement elasto-plastic analysis. However, it provides a reasonable upper bound load very economically, and with reasonable accuracy. It will be argued that it is pointless striving for a 1% accuracy when material parameters and load target values themselves can only be accurate to, say, 10%. The study has used software developed by Meek [3DBUCKLE-eigenvalue analysis program for buckling of steel columns, University of Queensland (1973)], which uses eigenvalue analysis by iteration to determine the Euler (theoretical) value of the buckling load of a column structure. This program was adapted to include the possibility that, during buckling, a reduced modulus section develops at the central portion of the critical column length. The reduction in the second moment of area of the section over a given length ensures that the revised estimate of the failure load is less than that predicted by the Euler formula for the undamaged column. The program was used to predict the failure loads of structures previously tested in other research programs, and the results show reasonable agreement with experiment. The paper shows the potential of the method as a simple design tool.

A pseudo element method (PEM) is presented in this paper for solving the matrix with elementary operation arising from the static analysis of cyclic symmetric structures. Three kinds of pseudo elements are devised to consider the transformed terms in the stiffness matrix during the elementary operation. Thus, the overall stiffness matrix may not be assembled by using the frontal solution technique. This can need relatively small computer memory and relatively short CPU time. The centrifugal displacements and stresses of a ring structure are calculated by introducing the cyclic symmetric conditions. The calculated results show, in comparison with theoretical ones, that the present method and its program are effective and can be conveniently used on a personal computer. At the end of this paper, the centrifugal fillet stresses of left-handed helical gear structures with different rim and web sizes are evaluated.

The present paper describes a novel method of generating a stiffness matrix for the elastic stress analysis of a uniform thin circular cylinder subject to bending and inplane loads. The method is based upon an exact solution of the equations of elasticity governing mechanical behaviour of a thin cylinder under prescribed loads. Stiffness coefficients are obtained computationally. Advantages of the proposed method are reduction of CPU time owing to far fewer elements than are necessary in the conventional finite element procedure and achievement of higher solution accuracy. The paper also presents results of application of the proposed method to some selected problems.

This paper presents a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path, and the existence of distinct critical loads (i.e. no coupling between buckling modes occurs). The formulation developed is known as W-formulation, in which the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the presentation developed here, the only ingredients required to carry out the analysis are the strain-displacement and the constitutive matrices at the element level. The present formulation is compared with the so called V-formulation, in which the displacements refer to the unloaded state. It is shown that under the present assumptions of linear fundamental path, the advantages of the V-formulation are lost and both approaches are similar. An example of a circular plate under in-plane loading illustrates the procedures. Part II of this paper deals with the application to the post buckling analysis of plate assemblies made of composite materials.

An energy-based, systematic method for the coupling of the finite element (FE) method and the boundary integral equation (BIE) method is described in this paper. This method allows the use of the BIE method to represent those subdomains of a structure that are best suited to BIE method, and the use of the FE method to represent the rest of the structure. Different subdomains and their associated FE or BIE representations are coupled naturally through the total potential energy functional of the system. The associated discretized problem from the proposed method consists of a linear system of equations with a symmetric and blockwise banded matrix. As in regular FE methods, the derivation is started from the total potential energy principle. However, in those subdomains that are best suited to the BIE method, the exact solution of the associated free field equation is used to represent the true solution. The size and shape of each BIE subdomain, called “macro-element” in this paper, may be designed freely to meet various practical requirements concerning, for example, numerical efficiency, machine storage limitations, mesh generations, etc. Most significantly, unlike the existing BIE techniques, the present method does not seem to require special treatments for corner effects, thus reducing the computational complexity. Numerical experiments have been performed for generalized Poisson's equation as a prototype situation. The extension to 2D-3D elastic problems is straightforward.

In a companion paper the authors presented a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path and the existence of distinct critical loads. The formulation developed is known as the W-formulation, where the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the present paper a number of applications of finite elements for post-buckling analysis on composite plate assemblies are presented. Thin-walled composite plates, I-beams, angle sections, and a specially designed box-beam with flanges (unicolumn) are studied in post-buckling when axially loaded. The results are in good agreement with previous studies. Moreover, a parametric study involving critical buckling load and geometry is presented for the case of the unicolumn.

In a companion paper [M. L. Liu and C. W. S. To, Comput. Struct. 54, 1031–1056 (1995)] theories and incremental formulation of nonlinear shell structures discretized by the finite element method are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Based on the theory and incremental formulation explicit element stiffness and mass matrices of three node flat triangular shell finite elements are derived. In the present paper the derived element matrices are applied to nine examples. The latter include static and dynamic response analysis of shell structures with geometrical, material, and geometrical and material nonlinearities. The formulation adopted and element matrices derived are found to be accurate, flexible and applicable to various types of shell structures with geometrical and material nonlinearities.

An efficient nonlinear approach is presented in this paper in matrix form for the analysis of the static and dynamic response of steel frames subjected to horizontal loads arising from wind or earthquakes. This approach is particularly suitable for developing a computer program for evaluating ultimate capacities of buildings in steel frames. The stiffness matrix equations for basic elements of steel frames are first outlined in the paper. The effects of geometric nonlinearity, material nonlinearity and shear deformation are considered in the proposed stiffness matrices for beam and column elements. Next, the stiffness equations of the extended column elements with panels attached and extended hybrid beam elements attached to both connections and panels are established to consider the effects of shear deformation of panels and flexibility of beam-to-column connections. Then the nonlinear behaviour of an arbitrary steel frame with flexible connections and deformable panels can be analysed by simply assembling all the stiffness matrix equations of the panel elements, extended column elements and extended hybrid beam elements of the frame to form the overall stiffness equation of the entire frame. Finally, the robustness, accuracy and efficiency of the proposed approach are demonstrated through several numerical examples.

A simple four-noded geometrically nonlinear shell element, which handles arbitrarily large displacements and rotations, is presented in this paper. Based on the assumption of small incremental strain in each load step, a corotational procedure is employed to extract the pure deformational displacements and rotations and update element stresses and internal force vectors through a piece-wise linearized strain-displacement relation. To derive the tangent stiffness matrix, a three-dimensional degenerate, isoparametric shell model is employed. The ‘locking’ problem is alleviated by using mixed interpolation of tensorial transverse shear strain components. The approach described in this paper is ideally suited for implementation in existing linear finite element programs. A number of numerical examples are also presented.

This paper is concerned with the structural optimization problem of minimizing the weight of a two-bar truss subject to nonlinear stability constraints to result in a degree of complexity beyond that which is expected for problems of this size. Optimization problems for stability constraints are commonly handled as generalized eigenvalue problems with the resulting designs overestimating true buckling loads by factors of up to 7.55. This paper presents an exact formulation that includes conditions for instability and optimality conditions in addition to optimal designs for symmetric cases and nonsymmetric cases using nonlinear programming. Multimember trusses are further considered and an exact displacement method of analysis developed.

This paper presents a weighted p-version least squares finite element formulation (LSFEF) for two-dimensional plane elasto-statics and its application for simulating linear static behavior of laminated composites. The second order partial differential equations of equilibrium in displacements are recast into a series of first order coupled partial differential equations using stresses as auxiliary variables. The primary (displacements: u and v) and the secondary (stresses: σxx, τxy and σyy) variables are interpolated using equal order, C0 continuity, p-version hierarchical approximation functions. The p-version hierarchical approximation functions and the corresponding nodal variable operators are derived directly from the Lagrange family of interpolation functions by first constructing the one dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators for three node equivalent configurations in the ξ and η directions and then taking their products. The least squares procedure yields a system of linear, symmetric simultaneous algebraic equations. The formulation provides inter-element continuity of displacements as well as stresses. Even though the element approximation is of type C0, in the limit when the error functional I converges to zero (i.e., when I → 0) the formulation behaves more like a C1 type. In addition, for an inadequate mesh with an inadequate order of approximation for the elements, the element error functional values provide a mechanism for h and/or p-adaptive refinements. This is a unique feature of the LSFEF as compared to variational based formulations. Many other unique features of the formulation are discussed and illustrated in the paper. A numerical example is presented to illustrate the effectiveness of the formulation in simulating accurate inter-lamina stress behavior as well as accurate stress values and stress gradients at and near singularities. The results obtrained from the present formulation are compared with the results obtained from the p-version variational formulation and with those reported in the literature.

This paper deals with the dynamical problem of ships side launching, which is an advanced technique in ship-building. A series of mechanical and mathematical models have been conducted with the ship side launching, in which the motion of a side launching ship was three degrees of freedom and the motion procedures were four degrees of freedom. The paper presents the solution of theoretical analysis in the form of numerical methods and a computer program to apply to ship-building conveniently. A practical example of ship side launching is shown in the paper.

This paper deals with the free vibration analysis of shells of revolution using the finite element method. A family of variable thickness, curved C (0) Mindlin-Reissner axisymmetric finite elements is presented which include shear deformation and rotatory inertia effects. The accuracy, convergence and efficiency of these newly developed elements are explored through a series of free vibration analyses of axisymmetric shell structures and the results are compared with those obtained by other analytical and numerical methods. The comparisons show that the method yields very good results with a relatively small number of elements and that estimates for the higher modes can be obtained without any difficulties. A companion paper will consider the structural optimisation of axisymmetric shells undergoing free vibrations.

This second part of the paper deals with the development and application of computational tools for structural shape optimization of prismatic folded plates and shells with curved planform. The structural analysis is carried out using the finite strip method formulated and tested in part I of this paper. Several examples of prismatic folded plates and shells are presented illustrating optimal shapes obtained by minimizing weight or strain energy. The changes in the relative contributions of the bending, membrane and shear strain energies are also monitored during the whole process of optimization to study energy distribution in optimum structures.

processes when analyzing their thermal behavior. The finite element formulation for such a problem on a process domain produces a system of ordinary differerential equations which is characterized to be ‘stiff. Careful handling of the solution is required in order to avoid oscillatory and inaccurate results. This problem can be avoided by selecting a suitable recurrence scheme together with an efficient steplength control algorithm. In this paper an automatic steplength control algorithm has been developed and implemented in a finite element program for the transient convective-transport equation. The algorithm has been tested using an example representing a manufacturing process. The conv~tiv~trans~rt equation is applied to many manufacturing processes to determine the transient temperature distribution [l-7]. It is applied to metal cutting, metal casting, forming, welding, etc. The analytical solution is found to be difficult and often not possible in such applications due to the complexity of the problem [4]. Therefore, numerical methods such as finite difference and finite element are usually employed. The finite element fo~ulation produces a system of first-order ordinary differential equations (ODES). The system of ODES is found to be, in most cases, very ‘stiff’ and produces oscillatory results when the time step is inadequate at the initial stages of the solution [8,9]. In order to avoid these problems, there is a need for an automatic time steplength control algorithm which can be used to automatically determine the time steps for a stable, accurate and economic solution [IO]. This paper presents a new automatic steplength control algorithm that is used with suitable recurrence schemes in solving such systems of ODES. It also gives a discussion for the formulation of the problem, and the selection of the recurrence schemes.

In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler-Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corresponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facilitating numerical models that are not only reliable but also computationally efficient. The authors acknowledge support from MCIN/ AEI/10.13039/501100011033 under Grants numbers PGC2018-098218-B-I00 and PRE2019-088002. FEDER: A way to make Europe. ESF invests in your future.

In this paper, a new paradigm for Carrera’s Unified Formulation (CUF) based on multiscale structural modelling is accomplished by bridging micromechanics and the advanced CUF one-dimensional/beam structural theories by means of the Multilevel Finite Element (also known as FE2) framework. Under the framework of the FE2 method, the analysis is divided into a macroscopic/structural problem and a microscopic/material problem. At the macroscopic level, several higher-order refined beam elements can be easily implemented via CUF by deriving a fundamental nucleus that is independent of the approximation order over the thickness and the number of nodes per element (they are free parameters of the formulation). The unknown macroscopic constitutive law is obtained by numerical homogenisation of a Representative Volume Element (RVE) at the microscopic level. Vice versa, the microscopic deformation gradient is calculated from the macroscopic model. Information is passed between the two scales in a FE2 sense. The resulting nonlinear problem is solved through the Asymptotic Numerical Method (ANM) that is more reliable and less Newton-Raphson’s one. The developed models are used as a first attempt to investigate the microstructure effect on the macrostructure geometrically nonlinear response. The proposed paradigm is used for investigating the effect of microscale imperfections (not straight carbon fibres) on the macroscale response (instability). Results are assessed in terms of accuracy and computational costs towards full FEM solutions. Three factors have been considered for an imperfection sensitivity parametric analysis: the defect wavelength as well as the amplitude and the size of RVE.

In this paper, a new numerical method, named as the Free Element Collocation Method (FECM), is proposed for solving general engineering problems governed by the second order partial differential equations (PDEs). The method belongs to the group of the collocation method, but the spatial partial derivatives of physical quantities are computed based on the isoparametric elements as used in FEM. The key point of the method is that the isoparametric elements used can be freely formed by the nodes around the collocation node. To achieve a narrow bandwidth of the final system of equations, elements with a central node are recommended. For this purpose, a new 21-node quadratic element for 3D problems is constructed for the first time. Attributed to the use of isoparametric elements, FECM can result in more stable results than the traditional collocation method. In addition, the elements can be freely formed by local nodes, FECM has the advantage of mesh-free methods to fit complicated geometries of engineering problems. A number of numerical examples of 2D and 3D thermal and mechanical problems are given to demonstrate the correctness and efficiency of the proposed method.

The paper presents a hybrid numerical method for the computation of the dispersion characteristics of ultrasonic guided waves propagating in isotropic and viscoelastic thin-walled waveguides in contact with fluids along their inner or outer wall. To this end, the solid waveguide is modeled by means of a Semi-Analytical Finite Element method implementing the first order shear deformation theory, while the effects of the fluid are taken into account by means of a regularized 2.5D Boundary Element Method with Helmholtz-type kernels. By coupling the two methods along the mid-surface of the solid waveguide, the dispersion relations of the elastic-acoustic system are obtained as the solution of a nonlinear eigenvalue problem in the complex axial wavenumbers for any fixed circular frequency. Two different case studies are presented: an oil-filled pipe and a water-loaded steel rectangular waveguide.

To study the magneto-fluid-structure interaction (MFSI) problems for rigid bodies and conducting fluids, a numerical method and an analytical approach have been carried out. The numerical scheme is based on a partitioned arbitrary Lagrangian-Eulerian framework, and is suitable for viscous, incompressible magneto-fluid-structure interaction simulation. A displacement prediction-pressure stabilization scheme has been established to enhance the stability and efficiency. Meanwhile, a consistent and conservative scheme for deforming configurations has been developed. This method can numerically ensure the divergence-free condition of the current density, and can conserve the momentum from the Lorentz forces after grids update. The analytical approach has considered a vibrating cylinder surrounded by confined fluids in a magnetic field. By assuming a small amplitude and a low magnetic Reynolds number, the analytical solution can describe the temporal and spatial distribution of the fluid fields, the electromagnetic fields, and the solid motion. These solutions are also suitable for general fluid-structure interaction (FSI) problems. Comparative results suggest good agreement between the two methods developed in this paper. Nonlinear effects of the magnetic fields were presented and discussed based on the numerical results. These cases are based on careful validations, and can hopefully be used for future verification and validation work.

In this paper, a non-standard finite element (FE) method, coupled with a mathematical programming procedure, is presented to simulate initiation and propagation of in-plane tensile cracks in quasi-brittle material. Potential crack lines are defined and limited by the FE edges. The cracks are considered as localized plastic deformations imposed at the nodes along the potential crack paths. Crack opening is detected at each node by a limit value of tensile force normal to the crack direction. When the limit value is reached, a cohesive crack starts, following a linear softening branch. The mechanical model is such that no remeshing neither the introduction of interfaces is necessary to describe the discontinuity between elements, as usually required in standard FE approaches. The non-linear solution of the structural problem is obtained using a mathematical programming procedure based on the solution of a parametric linear complementarity problem (PLCP). The main advantage of the PLCP formulation is its ability to manage structural configurations in which instability and multiplicity of solutions are possible, providing one solution even at points characterized by a not-positive-semidefinite Hessian matrix. Simple numerical simulations are presented with the aim to test the method and the solution procedures.

Mathematical models for the optimization problems of metal structures at shakedown definitely contains strength (ULS – ultimate limit state) and stiffness (SLS – serviceability limit state) constraints. Residual displacements determined by plastic deformations appear in stiffness constraints. The residual displacements developing during shakedown process of ideally elastic-plastic structures under variable repeated load can vary non-monotonically. Therefore, it is important to determine variation bounds of residual displacements (particularly in the cases when only variation bounds of loading rather than a particular history of a load are known). Thus, numerical methods, extremum energy principles, the assumption of a small displacement and mathematical programming theory have been used in the paper to develop mathematical models for the analysis problem of residual displacements. Using the proposed scanning method, it is acceptable to define the revised bounds of residual displacements without analysing the detailed history of loading. However, scanning technique is approximate method comparing by comprehensive analysis of elastic-plastic structure, when to exact values of displacement are obtained. The application of the proposed scanning method is illustrated by a benchmark of a plane multi-supported beam.

In this paper, a novel 3D detailed micro-model to analyse the mechanical response of masonry panels under in-plane and out-of-plane loading conditions is proposed. The modelling approach is characterized by textured units, consisting of one brick and few mortar layers, represented by 3D solid finite elements obeying to plastic-damage constitutive laws. Textured units are assembled, accounting for any actual 3D through-thickness arrangement of masonry, by means of zero-thickness rigid-cohesive-frictional interfaces, based on the contact penalty method and governed by a Mohr-Coulomb failure surface with tension cut-off. This novel approach can be fully characterized by the properties obtained on small-scale experimental tests on brick and mortar and on small masonry assemblages. The interface behaviour appears consistent with small-scale tests outcomes on masonry specimens. Experimental-numerical comparisons are provided for the in-plane and out-of-plane behaviour of masonry panels. The accuracy, the potentialities and the efficiency of the modelling approach are shown and discussed.

This paper presents a Dynamic Partitioning Method (DPM) to solve the vehicle-bridge interaction (VBI) problem via a set of exclusively second-order ordinary differential equations (ODEs). The partitioning of the coupled VBI problem follows a localized Lagrange multipliers approach that introduces auxiliary contact bodies between the vehicle’s wheels and the sustaining bridge. The introduction of contact bodies, instead of merely static points, allows the assignment of proper mass, damping and stiffness properties to the involved constrains. These properties are estimated in a systematic manner, based on a consistent application of Newton’s law of motion to mechanical systems subjected to bilateral constraints. In turn, this leads to a dynamic representation of motion constraints and associated Lagrange multipliers. Subsequently, both equations of motion and constraint equations yield a set of ODEs. This ODE formulation avoids constraint drifts and instabilities associated with differential–algebraic equations, typically adopted to solve constrained mechanical problems. Numerical applications show that, when combined with appropriate numerical analysis schemes, DPM can considerably decrease the computational cost of the analysis, especially for large vehicle-bridge systems. Thus, compared to existing methods to treat VBI, DPM is both accurate and cost-efficient.

A comparative study of three composite implicit schemes including the Bathe scheme (Bathe and Baig, 2005), the TTBDF scheme (Chandra et al., 2015) and the Wen scheme (Wen et al., 2017) is conducted in this paper. The stability and accuracy characteristics of these schemes are studied and compared by analytical and numerical simulation analysis. In the solution of wave propagation problems, a demonstrative dispersion analysis is given and problems are solved to illustrate the capabilities of the schemes for the solution of wave propagation problems as well as the numerical dissipation characteristics of the schemes. The nonlinear dynamic behavior of the Wen scheme is studied by considering commonly used nonlinear dynamic problems where the nonlinear performance of the scheme is exclusively compared with the Bathe and TTBDF schemes. The priority ranks of three presented composite schemes for different types of dynamic problems are obtained by theoretical and numerical analysis.

This paper presents a numerical method to control the spurious deformation mode conventionally found in a non-ordinary state-based peridynamic formulation. The proposed approach introduces a higher-order approximation for a deformation gradient tensor in order to suppress oscillations from the zero-energy mode. The results are compared to those determined by other available methods. The findings of this study well demonstrate that the proposed method provides highly effective solutions for controlling zero-energy modes in peridynamic models. Contrasting other available zero-energy control methods, the proposed method remains robust for relatively large horizon sizes. The method is incorporated by implicit weight functions in peridynamics.

Dynamic fracturing of rock masses simulated by the conventional FEM, XFEM and the element deletion method.XFEM analysis of dynamic fracturing with and without pre-existing cracks.FEM and XFEM analysis of dynamic fracturing with and without impenetrability constraint.Cohesive fractures modeled by the method of phantom nodes. This paper presents a numerical analysis based on the finite element method to simulate blast-induced hard rock fracture propagation. Three different approaches are compared: the extended finite element method, with a cohesive zone model to represent the growth of fractures; the conventional finite element method using a remeshing technique and based on the linear fracture mechanics; the element deletion method to simulate a rock fragmentation process. The rock mass is a sound granite that remains linear elastic right up the breakage. Two numerical examples are presented in order to discuss the advantages and limitations of each approach.

An efficient and accurate method for buckling analysis of axially loaded periodic cylindrical shells is presented.This method incorporates the advantages of the Bloch wave method and the stiffness matrix method.This method does not require any smearing, homogenization, or approximation.This method does not require specific-purpose elements and is developed in the environment of Abaqus. This paper presents an efficient computational method for predicting the onset of buckling of axially loaded, corrugated or stiffened cylindrical shells. This method is a modification of the Bloch wave method which builds on the stiffness matrix method. A numerical method and an efficient algorithm have been developed to implement the proposed method in the commercial finite element package Abaqus. Numerical examples have shown that, compared to the nonlinear buckling analyses based on detailed full finite element models, the proposed method can obtain highly accurate buckling loads and buckling modes and can achieve very significant reductions in computational time.

A number of physical systems can be described by ordinary differential equations. When physics is well understood, the time dependent responses are easily obtained numerically. The particular numerical method used for integration depends on the application. Unfortunately, when physics is not fully understood, the discrepancies between predictions and observed responses can be large and unacceptable. In this paper, we propose an approach that uses observed data to estimate the missing physics in the original model (i.e., model-form uncertainty). In our approach, we first design recurrent neural networks to perform numerical integration of the ordinary differential equations. Then, we implement the recurrent neural network as a directed graph. This way, the nodes in the graph represent the physics-informed kernels found in the ordinary differential equations. We quantify the missing physics by carefully introducing data-driven in the directed graph. This allows us to estimate the missing physics (discrepancy term) even for hidden nodes of the graph. We studied the performance of our proposed approach with the aid of three case studies (fatigue crack growth, corrosion-fatigue crack growth, and bearing fatigue) and state-of-the-art machine learning software packages. Our results demonstrate the ability to perform estimation of discrepancy, reducing gap between predictions and observations, at reasonable computational cost.

Spectral element method (SEM) is a robust and efficient mathematical technique for dynamic analysis of structures in frequency domain. Unlike finite element method (FEM), in SEM, the dynamic stiffness matrix forms a nonlinear eigenvalue problem (NLEP) to compute the natural frequencies and vibration modes of the structure which cannot be solved using linear numerical eigen-solvers. In this paper, two distinct numerical methods, i.e. (1) a root finding method of rational polynomial functions and (2) a linearization of Lagrange matrix interpolating polynomials, have been used to compute the eigenvalues of a problem more efficiently employing SEM. These proposed methods can solve NLEP in a stable, efficient and accurate way even in the presence of singularities. The accuracy of these methods are numerically evaluated by comparing with the solutions from the modal analysis using FEM.