MITC elements for a classical shell model

The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the mixed interpolation of tensorial components approach (MITC), but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem––the Scordelis-Lo roof.     

Improving the computational efficiency in finite element analysis of shells with uncertain properties

This work is concerned with improving the computational efficiency of the most time consuming tasks performed in Monte Carlo simulation-based Finite Element Analysis (FEA) of shell structures with uncertain properties. For this purpose, stochastic field values are generated on a coarse mesh and then interpolated onto the fine mesh used for the standard FEA computations; the cost-effective TRIC shell element is used to ensure the formation of stiffness matrices in reasonable processing times; the solution of finite element equations is efficiently handled with hybrid schemes combining both iterative and direct solution concepts; additional computational gains are achieved with the use of parallel computing through the straightforward partitioning of the overall Monte Carlo simulation process. The adoption of such advanced computational approaches allows simulation-based probabilistic or stochastic FEA of shells to be performed in affordable computing times and therefore become more tractable in structural engineering practice. The computational procedures described in this work are evaluated on a cluster of 16 networked PCs using three linear elastic test problems with uncertain material and/or geometric parameters: (a) the Scordelis-Lo shell, (b) a pinched cylinder and (c) a 3D steel frame discretized with shell elements.     

A multi-director formulation for nonlinear elastic-viscoelastic layered shells

A C⁰ finite element formulation for nonlinear analysis of multi-layered shells comprised of elastic and viscoelastic layers is presented for applications involving small strains but finite rotations. The elastic and viscoelastic layers may occupy arbitrary layer locations and the formulation is applicable to thick and thin shells. The formulation utilizes a three-dimensional variational approach in which the layered shell is represented as a multi-director field. The incorporated kinematic theory describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness coordinate. Stresses are computed through the three-dimensional constitutive equations and the usual “zero normal stress” shell hypothesis is not employed. Sufficiently general constitutive equations for the viscoelastic layers are proposed in objective rate form and a product algorithm, based on an operator split in the complete set of constitutive equations, is used for the temporal integration of the rate equations. The definition of the tangent operator, used in Newton's method for the solution of the nonlinear equations, is derived consistently from the product algorithm. Observations on the use of reduced/selective integration in the presence of high order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.     

Omnidirectional texturing based on robust 3D registration through Euclidean reconstruction from two spherical images

We propose a semi-automatic omnidirectional texturing method that maps a spherical image onto a dense 3D model obtained by a range sensor. For accurate texturing, accurate estimation of the extrinsic parameters is inevitable. In order to estimate these parameters, we propose a robust 3D registration-based method between a dense range data set and a sparse spherical image stereo data set. For measuring the distances between the two data sets, we introduce generalized distances taking account of 3D error distributions of the stereo data. To reconstruct 3D models by images, we use two spherical images taken at arbitrary positions in arbitrary poses. Then, we propose a novel rectification method for spherical images that is derived from E matrix and facilitates the estimation of the disparities. The experimental results show that the proposed method can map the spherical image onto the dense 3D models effectively and accurately.     

Resultant-stress degenerated-shell element

In this paper, a resultant-stress degenerated-shell element is described and a variety of numerical examples, including the post-buckling analysis of an axially loaded perfect cylinder, are presented. The general degenerated nonlinear shell theory of Hughes and Liu is employed in deriving this resultant-stress degenerated-shell element.Contrary to the traditional integration through the thickness approach, which assumes no coupling between the in-plane and transverse material and structural response matrices, the present approach can permit use of arbitrary, three-dimensional (3-D) nonlinear constitutive equations. Furthermore, explicit expressions of the element matrices for a 4-node shell element are developed. This rank-sufficient 4-node shell element, termed the resultant-stress degenerated-shell (RSDS) element, avoids the need for the costly numerical quadrature function evaluations of the element matrices and force vectors. And thus there are large increases in computational efficiency with this method. The comparisons of this RSDS element with six other shell elements are also given in this paper.     

Large displacements of shells including material nonlinearities

The paper presents a finite element formulation of shells with large deflections including elastoplastic material behaviour. The elements utilized are the so-called degenerated shell elements, with special emphasis on the simple 4 node quadrilateral. Both the total Lagrangian and the updated Lagrangian formulation are considered. For the treatment of plastic behaviour the concept of a layered element model is proposed and investigated for both the tangential modulus method and the initial load method. In the final section sample problems are presented and compared with reference solutions. It is shown that the 4 node element is very well-suited to the class of problems under consideration. It is characterized by an easy applicability, high accuracy and low rates of computer time.In Appendix A an analytical solution of a cantilever beam, subjected to an end load, including geometrical and material nonlinearities is presented.     

Finite element analysis of reinforced and prestressed concrete structures including thermal loading

This paper describes the application of finite element techniques to the solution of nonlinear concrete problems. Reinforced concrete thick plates and shells are first considered for which both a perfect and strain-hardening plasticity approach are employed to model the compressive behaviour. A dual criterion for yielding and crushing in terms of stresses and strains is considered, which is complemented with a tension cut-off representation. Degenerate thick shell elements employing a layered discretisation through the thickness are adopted and both reduced and selectively integrated 8-node serendipity and heterosis elements are considered.Thermal loading of prestressed concrete structures is also considered which necessitates the inclusion of time effects in the analysis. The technique described in this paper involves concurrently solving an uncoupled set of equations within a time interval to provide both the displacement and temperature increments. A two-level time stepping scheme is employed to predict temperature changes within a time interval and elasto-viscoplastic material analysis is performed using an explicit forward-difference scheme incorporating an equilibrium iteration procedure. The constitutive model for the concrete is essentially identical to that employed for the plate and shell analysis.Numerical examples are presented for both types of analysis and comparison is made with experimental results whenever possible. Additionally, results for thermal loading are presented which indicate that a full transient thermal-mechanical analysis is sometimes essential in order to obtain a realistic structural response.     

An integrated framework for multi-criteria optimization of thin concrete shells at early design stages

Thin shells are crucially dependent on their shape in order to obtain proper structural performance. In this context, the optimal shape will guarantee performance and safety requirements, while minimizing the use of materials, as well as construction/maintenance costs.Thin shell design is a team-based, multidisciplinary, and iterative process, which requires a high level of interaction between the various parties involved, especially between the Architecture and Engineering teams. As a result of technological development, novel concepts and tools become available to support this process. On the one hand, concepts like Integrated Project Delivery (IPD) show the potential to have a high impact on multidisciplinary environments such as the one in question, supporting the early decision-making process with the availability of as much information as possible. On the other hand, optimization techniques and tools should be highlighted, as they fit the needs and requirements of both the shell shape definition process and the IPD concept. These can be used not only to support advanced design stages, but also to facilitate the initial formulation of shape during the early interactions between architect and structural engineer from an IPD point of view.This paper proposes a methodology aimed at enhancing the interactive and iterative process associated with the early stages of thin shell design, supported by an integrated framework. The latter is based on several tools, namely Rhinoceros 3D, Grasshopper, and Robot Structural Analysis. In order to achieve full integration of the support tools, a custom devised module was developed, so as to allow interoperability between Grasshopper and Robot Structural Analysis. The system resorts to various technologies targeted at improving the shell shape definition process, such as formfinding techniques, parametric and generative models, as well as shape optimization techniques that leverage on multi criteria evolutionary algorithms. The proposed framework is implemented in a set of fictitious scenarios, in which the best thin reinforced concrete shell structures are sought according to given design requirements. Results stemming from this implementation emphasize its interoperability, flexibility, and capability to promote interaction between the elements of the design team, ultimately outputting a set of diverse and creative shell shapes, and thus supporting the pre-design process.     

Nearest neighbor weighted average customization for modeling faces

In this paper, we present an anatomically accurate generic wireframe face model and an efficient customization method for modeling human faces. We use a single 2D image for customization of the generic model. We employ perspective projection to estimate 3D coordinates of the 2D facial landmarks in the image. The non-landmark vertices of the 3D model are shifted using the translations of k nearest landmark vertices, inversely weighted by the square of their distances. We demonstrate on Photoface and Bosphorus 3D face data sets that the proposed method achieves substantially low relative error values with modest time complexity.     

The quadratic MITC plate and MITC shell elements in plate bending

The analysis of plates can be achieved using the quadratic MITC plate or MITC shell elements. The plate elements have a strong mathematical basis and have been shown to be optimal in their convergence behavior, theoretically and numerically. The shell elements have not (yet) been analyzed mathematically in depth for their rates of convergence, with the plate/shell thickness varying, but have been shown numerically to perform well. Since the shell elements are general and can be used for linear and nonlinear analyses of plates and shells, it is important to identify the differences in the performance of these elements when compared to the plate elements. We briefly review the quadratic quadrilateral and triangular MITC plate and shell elements and study their performances in linear plate analyses.     

Logic programs with abstract constraint atoms: The role of computations

We provide a new perspective on the semantics of logic programs with arbitrary abstract constraints. To this end, we introduce several notions of computation. We use the results of computations to specify answer sets of programs with constraints. We present the rationale behind the classes of computations we consider, and discuss the relationships among them. We also discuss the relationships among the corresponding concepts of answer sets. One of those concepts has several compelling characterizations and properties, and we propose it as the correct generalization of the answer-set semantics to the case of programs with arbitrary constraints. We show that several other notions of an answer set proposed in the literature for programs with constraints can be obtained within our framework as the results of appropriately selected classes of computations.     

Evaluation of modeling accuracy of 20-node solid elements by statistical factorial design

Many complex engineering structures consist of plates and shells. When applying finite element methods to a plate or shell, how can a design engineer choose an initial mesh and decide the integration order? The statistical design of experiments is employed to investigate the influence of three relevant modeling factors—aspect ratio, number of elements per edge of model and integration order on the accuracy of applying 20-node solid elements to plates and shells. In addition to the main effects of individual factors, the effects of multiple-factor interactions among factors are addressed. Given a certain aspect ratio and integration order for a structure under a specific load, a minimum number of elements per edge of model can be predicted with a given statistical confidence coefficient in modeling.     

A family of special-purpose elements for analysis of ribbed and reinforced shells

A family of super-parametric special-purpose finite elements for analysis of ribbed and reinforced concrete shells is introduced. Any shell element may comprise an arbitrary number of curved ribs and/or reinforcing bars. The finite element formulation is conceived as an extension of Ahmad's thick shell element. The displacements and deformations of the ribs and/or reinforcing bars are consequently derived from the customary displacement definition of the thick shell elements. The formulation properly takes into account the excentricity of the ribs and/or reinforcing bars with respect to the middle surface of the plate or shell. Examples shown at the end of the paper illustrate the great efficiency of the concept in practical applications.     

Effect of boundary conditions on natural frequencies for rotating composite cylindrical shells with orthogonal stiffeners

The effect of the boundary conditions on the natural frequencies for rotating composite cylindrical shells with the orthogonal stiffeners is investigated using Love’s shell theory and the discrete stiffener theory. The frequency equation is derived using the Rayleigh–Ritz procedure based on the energy method. The considered boundary conditions are four sets, namely: (1) clamped–clamped; (2) clamped–simply supported; (3) clamped–sliding; and (4) clamped–free. The beam modal function is used for the axial vibration mode and the trigonometric functions are used for the circumferential vibration mode. The composite shells are stiffened with uniform intervals and the stiffeners have the same material. By comparison with the previously published analytical results for the rotating composite shell without stiffeners and the orthogonally stiffened isotropic cylindrical shells, it is shown that natural frequencies can be determined with adequate precision.     

3-D limit load analysis of reinforced concrete shells of arbitrary shape

A limit analysis method for thick reinforced concrete shells of arbitrary shape is developed using a 3-D concrete model based on a Mohr-Coulomb fracture theory in a solid-like isoparametric element. The proposed approach is well suited to engineering requirements as is illustrated by a HP shell case study     

A goal-oriented hp-adaptive finite element approach to radar scattering problems

The paper presents application of an hp-adaptive finite element method for scattering of electromagnetic waves. The main objective of the numerical analysis is to determine the characteristics of the scattered waves indicating the power being scattered at a given direction––i.e. the radar cross-section (RCS). This is achieved considering the scattered far-field which defines RCS and which is expressed as a linear functional of the solution. Techniques of error estimation for the far-field are considered and an h-adaptive strategy leading to the fast reduction of the error of the far-field is presented. The simulations are performed with a three-dimensional version of an hp-adaptive finite element method for electromagnetics based on the hexahedral edge elements combined with infinite elements for modeling the unbounded space surrounding the scattering object.     

Time‐Discrete Geodesics in the Space of Shells

Building on concepts from continuum mechanics, we offer a computational model for geodesics in the space of thin shells, with a metric that reflects viscous dissipation required to physically deform a thin shell. Different from previous work, we incorporate bending contributions into our deformation energy on top of membrane distortion terms in order to obtain a physically sound notion of distance between shells, which does not require additional smoothing. Our bending energy formulation depends on the so‐called relative Weingarten map, for which we provide a discrete analogue based on principles of discrete differential geometry. Our computational results emphasize the strong impact of physical parameters on the evolution of a shell shape along a geodesic path.     

Numerical analysis of anisotropic rotational shells subjected to nonsymmetric loads

Through the use of an integral transform, the present paper extends the applicability of the multisegment numerical integration technique to include the solution of general macroscopically anisotropic multilayered shells of revolution. It is found that compared with orthotropic shells, material anisotropy induces a doubling in the number of transformed fundamental equations characterizing the static response. Employing the transform together with the multisegment integration technique and several concepts from the direct stiffness method of structural analysis, procedures are developed which can handle branched anisotropic multilayered shells of revolution in a more effective manner than was previously possible. Based on the procedures outlined in the paper, numerical studies are presented which show the effect of segment size on the solution accuracy and the effects of material anisotropy on selected shell configurations.     

A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections

In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.