Element Displacement Field Research Articles

A diffusive‐discrete crack transition scheme for ductile fracture at finite strain

SummaryThis study presents a diffusive‐discrete crack transition scheme for stably conducting ductile fracture simulations within a finite strain framework. In the developed scheme, the crack initiation and propagation processes are determined according to an energy minimization problem based on crack phase‐field theory, and the predicted diffusive path is transformed to a discrete representation using the finite cover method during the staggered iterative scheme. In particular, for stably conducting ductile fracture simulations, three computational techniques, the staggered iterative configuration update technique, the subincremental damage update technique, and the crack opening stabilization technique, are introduced. The first and second techniques can be used regardless of whether discrete cracks are considered, and the third technique is specialized for diffusive‐discrete crack transition schemes. Accordingly, ductile fracture simulations with the developed scheme rarely encounter troublesome problems such as oscillations in the displacement field and severe distortion of finite elements that can lead to divergence of calculations. After the crack phase‐field model for ductile fractures is formulated and discretized, the numerical algorithms for realizing the diffusive‐discrete crack transition while maintaining computational stability are explained. Several representative numerical examples are presented to demonstrate the performance and capabilities of the developed approach.

A corotational isogeometric assumed natural strain shell element in updated lagrangian formulation for general geometric nonlinear analysis of thin-walled structures

A new general geometric nonlinear curved quadrilateral shell element, which can handle large displacements and large rotations, is presented in this paper. It is formulated based on the incremental virtual work equation in updated Lagrangian formulation for general large displacement analysis. The kinematics of shell element is described using non-uniform B-spline patch, which results in the element displacement field with higher order continuity and the element performance more resistive to shear locking and membrane locking. In order to perfectly avoid such membrane locking and shear locking behaviors, the assumed in-plane membrane strain field and the assumed transverse shear strain field for the incremental covariant Green-Lagrange strain are constructed as linear combinations of non-uniform B-spline basis functions with degrees lower than those adopted by the original displacement field interpolation. In addition, based on Euler parameters, the corotational procedure available for high order isogeometric shell element is developed to extract the strain-producing deformational displacements and rotations and to update the element stresses and realistic internal force vectors. The accuracy and robustness of proposed shell element are demonstrated through the solutions of various benchmark problems.

Study on Propagation Characteristics of Guided Waves in Plate Structure via Semianalytical Wavelet Finite Element Method

The propagation characteristics of guided wave in waveguide structure plays an indispensable role in guided wave inspection. However, the analytical matrix method fails to solve the problems of wave propagation in waveguides with complex cross-section. The finite element method (FEM) has high computational cost for guided wave propagation problems. To overcome those shortcomings, a novel semi-analytical wavelet finite element (SAWFE) method is presented in this paper to study the guided wave propagation characteristics of plate structures. With the advantage of multi-scale and multi-resolution, the B-spline wavelet on the interval (BSWI) scale functions are adopted to construct one-dimensional BSWI elements and form the shape function. The element displacement field represented by the coefficients of wavelet expansions is transformed from wavelet space to physical space through the corresponding transformation matrix. Then, the constructed one-dimensional BSWI elements are employed to discretize the cross-section of waveguide structure, meanwhile, the wave displacement field in the wave propagation direction is described in an analytical way as a harmonic exponential function. The propagation characteristics of guided wave in aluminum plate and composite laminate are studied by using the proposed method, and the results show that the SAWFE method has high accuracy and efficiency in calculating dispersion curves and analyzing wave structure. The simulation analysis and experimental validation results further verify the effectiveness of the proposed method.

A Triangular Shell Element Based on Higher-order Strains for the Analysis of Static and Free Vibration

This research paper proposes a new triangular cylindrical finite element for static and free vibration analysis of cylindrical structures. The formulation of the proposed element is based on deep shell theory and uses assumed strain functions instead of displacement functions. The assumed strain functions satisfy the compatibility equations. This finite element possesses only the five necessary degrees of freedom for each of the three corner nodes. The element's displacement field, which contains higher-order terms, satisfies the requirement of rigid-body displacement. The element's performance is evaluated using various numerical static and free vibration tests for cylindrical shell problems, including an analysis of the effect of shell openings on natural frequencies. The results of the developed element are evaluated in comparison with published analytical and numerical solutions. The new cylindrical element's formulation is straightforward. Compared to the degenerate nine-node shell element and other elements, the results of the present element have shown excellent accuracy and efficiency in predicting static and free vibration of curved structures. This element only requires the use of very coarse meshes to converge. In addition, the triangular shape of this element is more advantageous than the quadrilateral shape when the geometric domain of the structure is deformed or complicated. Doi: 10.28991/CEJ-2022-08-10-06 Full Text: PDF

Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method

This paper presents a novel higher-order C1 continuous Chebyshev spectral element method for the free vibration analysis of rectangular thin plates with corner and inner cutouts. To achieve C1 continuity at the interfaces of adjacent elements, the Hermite interpolation functions through nonuniformly distributed Chebyshev nodes are employed to construct the element displacement field. A Gauss–Lobatto type quadrature is used to efficiently evaluate the element matrices. Comparison with other well-established methods show that the proposed element has good convergence and accuracy. The influence of cutout size, location and eccentric ratio on the natural vibration characteristics of plates are investigated.

Shape-Sensing of Beam Elements Undergoing Material Nonlinearities.

The use of in situ strain measurements to reconstruct the deformed shape of structures is a key technology for real-time monitoring. A particularly promising, versatile and computationally efficient method is the inverse finite element method (iFEM), which can be used to reconstruct the displacement field of beam elements, plate and shell structures from some discrete strain measurements. The iFEM does not require the knowledge of the material properties. Nevertheless, it has always been applied to structures with linear material constitutive behavior. In the present work, advances are proposed to use the method also for concrete structures in civil engineering field such as bridges normally characterized by material nonlinearities due to the behavior of both steel and concrete. The effectiveness of iFEM, for simply supported reinforced concrete beam and continuous beams with load conditions that determine the yielding of reinforcing steel, is studied. In order to assess the influence on displacements and strains reconstructions, different measurement stations and mesh configurations are considered. Hybrid procedures employing iFEM analysis supported by bending moment-curvature relationship are proposed in case of lack of input data in plastic zones. The reliability of the results obtained is tested and commented on to highlight the effectiveness of the approach.

Experimental and numerical investigations of reinforced concrete columns confined internally with biaxial geogrids

Geogrids are geosynthetic materials that have been proven effective for stabilization and strengthening applications in infrastructure, such as: reinforcement and stabilization of pavement layers, reinforcement of embankments and soil walls, in addition to soil improvement. The main aim of this study is to investigate the feasibility of using biaxial geogrids as a replacement of ordinary transverse steel reinforcement in columns. Using geogrids as confining material in reinforced concrete columns offers advantages over steel stirrups in that they are less laborious to install and are more durable. The study presents results of experimental and analytical investigations of the behavior of small-scale reinforced concrete columns of 500 mm height subjected to monotonic uniaxial compressive loading. The main test variables include confining material (steel stirrups or geogrids), number of geogrids layers, and the aspect ratio (height to diameter) of the column. Strain gauges were installed on the geogrids to determine the displacement fields of the geogrid elements. Axial displacement and axial load were measured to assess the overall load-displacement behavior, stiffness, ultimate strength, and ductility of the confined column specimens. Analysis of the results revealed that the ductility and energy absorption capacity of concrete column specimens are substantially enhanced without any significant effect on the ultimate load capacity when internally confined by biaxial geogrids. Two distinct analytical models were developed to predict the load-displacement behavior, one for unconfined plain concrete specimens, and the other for columns with transverse reinforcement using biaxial geogrids. The models provided satisfactory predictions of the stress–strain response when the analytical and experimental results were compared.

Nonlinear augmented finite element method for arbitrary cracking in large deformation plates and shells

AbstractThis article presents a nonlinear augmented finite element method (N‐AFEM) for the analysis of arbitrary crack initiation and propagation in large deformation plates and shells. The FE formulations for plate/shell elements and a shell‐like cohesive zone element, both with explicit consideration of geometric nonlinearity, have been derived in detail. The geometrically nonlinear shell‐like cohesive element has the essential feature of 3D but with crack displacements directly extracted from midplane shell element nodes, which enables an accurate description of crack propagation in shells and plates under large deformation. Furthermore, a novel augmentation process that can explicitly account for the discontinuous displacement fields of cracked elements without the need of extra nodes or nodal DoFs has been develop based on a nonlinear Newton‐Raphson method. The numerical performance of the N‐AFEM in modeling a number of benchmark shell/plate fracture problems demonstrates that the method is efficient, accurate, and robust.

Three-dimensional stress analyses of complex laminated shells with a variable-kinematics continuum shell element

This paper presents a new variable-kinematics continuum shell (VKCS) element that can be used to model laminated shell structures with arbitrary geometry in a finite element (FE) setting. The novelty is in the implementation of variable-kinematics capability in a continuum shell formulation, using Carrera’s Unified Formulation (CUF). The resultant model has completely general geometric and kinematic descriptions. In the formulation, the geometrical representation is based on a numerical isoparametric map with no simplifying assumptions on the shell geometry; whereas the element displacement fields are written in terms of the Fundamental Nuclei according to CUF. In the variable-kinematics framework, the levels of hp– and p–refinements in the through-thickness and in-plane domains are free parameters that can be varied independently. By parametrically varying in-plane mesh densities and model kinematics, model settings with good trade-offs in computational cost and desired level of accuracy can be identified. In addition to the existing literature benchmarks, we include new 3D stress benchmarks for laminated shells with complex geometrical features, such as spatially varying curvatures, non-orthogonal coordinate lines and variable thicknesses. The higher-order models yield asymptotically correct three-dimensional stresses, even in regions near singularities, without requiring numerical artefacts nor stress recovery procedures. In terms of computational efficiency, the model variants utilising high p-level require fewer total degrees of freedom (dofs) compared with linear 3D finite element method (FEM) for convergence of the 3D stress field. In terms of wider applications, the compact formulation can allow for the same computer code and model mesh to be used across a wide range of analyses for complex shell structures that requires different model fidelity, with minimal inputs from the user.

Coupled Transverse and Axial Vibrations Including Warping Effect in Asymmetric Short Beams

AbstractTo model the displacement field of asymmetric one-dimensional structural elements, in which a transverse-bending motion couples with axial motion, a formulation with five displacement compo...

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.

Numerical Simulation of Temperature Distribution Field in Beam Bulk in the Simultaneous Presence of Heat Insulation, Heat Flux and Heat Exchange

Most of the bearing elements of electric power installations operate in the presence of local cold insulator, heat flow, and heat transfer. To ensure the smooth operation of these elements it is necessary to develop appropriate computational algorithms and methods that allow to investigate numerically the thermomechanical condition of bearing elements. Besides, obtained numerical results should have different precision and respectively describe such a complex exceptional nonlinear process under study. Under the established process for the length of the test rod bearing element temperature field, displacement field, elastic, thermal and thermoelastic stress components field will arise. To determine these decisions, taking into account natural conditions and maintenance, we should use the techniques oriented on the basic laws of conservation of energy [1, 2].

Wave motion analysis and modeling of membrane structures using the wavelet finite element method

In this study, we present an application of the B-spline wavelet on the interval element method to analyze the in-plane elastic wave motion in a two-dimensional membrane structure. In contrast to traditional polynomial interpolation in classical finite element methods, the scaling function at a certain scale is used to form the shape functions and to construct wavelet-based elements. Other numerical wavelet methods add the wavelets directly, whereas the element displacement field represented by the coefficients of wavelets expansions in the proposed method is transformed from wavelet space to physical space via the corresponding transformation matrix. Numerical experiments are presented to demonstrate the effects of in-plane wave propagation in intact/notched membranes, particularly the propagations of the primary wave, secondary wave, and Rayleigh wave. In order to ensure a comprehensive analysis of the problem, the responses of the membrane are simulated under broad-band and narrow-band excitation.

Hybrid-Trefftz formulation for analysis of anisotropic and symmetric laminated plates

In this paper, two novel and efficient elements are proposed for analysis of anisotropic and symmetric laminated thick plates based on hybrid-Trefftz formulation. In this approach, two independent displacement fields are defined for internal and boundary of the elements. The internal field is selected in such a manner that the governing equation of thick plates could be satisfied. At the first step, the governing differential equations of anisotropic and symmetric laminated thick plates are derived. Then, the analytical solution of these equations is considered. By using these analytical functions, the internal displacement field of proposed elements is achieved. Also, the boundary field is related to the nodal degrees of freedom by the boundary interpolation functions. For this aim, the interpolation functions of tow-node laminated Timoshenko beam element are calculated for boundary field. By utilizing these displacement fields and hybrid-Trefftz functional, a triangular element THT-7 and quadrilateral element QHT-11 are proposed. These elements have 9 and 12 degrees of freedom, respectively. The results of the numerical testes are proven the absence of the shear locking and high accuracy and efficiency of the proposed elements for analysis of anisotropic and symmetric laminated plates.

Crack detection in concrete gravity dams using a genetic algorithm

The main aim of this paper is to introduce an efficient procedure for crack detection in concrete gravity dams. A genetic algorithm and finite-element modelling are employed to perform the optimisation tasks. A two-dimensional cracked element is formulated for non-linear analysis. In this investigation, the kinematics of the discontinuous displacement field in a body with an internal discontinuity is utilised. Based on the suggested interpolation functions for the discrete segments, and also the element displacement field, the element stiffness matrix is calculated. Moreover, a genetic algorithm approach for crack identification is proposed, which can identify the location and magnitude of cracks in concrete gravity dams. By minimising the difference between the analytical responses given by the authors' element and the measured ones, the genetic algorithm identifies the crack. Finally, several numerical examples are analysed for the accuracy test and a few of them are presented here.

Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods

Two 4-node generalized conforming quadrilateral membrane elements with drilling DOF, named QAC4θand QAC4θM, were successfully developed. Two kinds of quadrilateral area coordinates are used together in the assumed displacement fields of the new elements, so that the related formulations are quite straightforward and will keep the order of the Cartesian coordinates unchangeable while the mesh is distorted. The drilling DOF is defined as the additional rigid rotation at the element nodes to avoid improper constraint. Both elements can pass the strict patch test and exhibit better performance than other similar models. In particular, they are both free of trapezoidal locking in MacNeal’s beam test and insensitive to various mesh distortions.

A Novel Hexahedral Interface Element for Nonlinear Crack Analysis

Existence of a crack in structures would lead to a sudden failure and damage. Establishing a precise analytical model for the cracked element would be a powerful tool to achieve the right answers in the analysis of the structure. The main aim of this article is to formulate a hexahedral interface element for use in nonlinear crack analysis. In this investigation, the kinematics of the discontinuous displacement field along with the virtual work principle, for a body with an internal discontinuity, is utilized. Based on the suggested interpolation functions for the discrete segments, and also the element displacement field, the element stiffness matrix is calculated. The proposed element can be used for modeling of the discrete cracks in three-dimensional problems, such as a concrete dam. Several numerical examples are analyzed for the accuracy test and a few of them are presented here. The results indicated that utilizing sufficient elements yields suitable answers.

The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method

A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.

Wave motion analysis in arch structures via wavelet finite element method

Abstract The application of B-spline wavelet on interval (BSWI) finite element method for wave motion analysis in arch structures is presented in this paper. Instead of traditional polynomial interpolation, scaling functions at certain scales have been adopted to form the shape functions and construct wavelet-based elements. Different from other wavelet numerical methods adding wavelets directly, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The energy functional of the arch is obtained by the generalized shell theory, and the finite element model for wave motion analysis is constructed according to Hamilton's principle and the central difference method in time domain. Taking the practical application into account, damaged arch waveguides are also investigated. Proper analysis of the responses from structure damages allows one to indicate the location very precisely. This paper mainly focuses on the crack in structures. Based on Castigliano's theorem and the Pairs equation, the local flexibility of crack is formulated for BSWI element. Numerical experiments are performed to study the effect of wave propagations in arch waveguides, that is, frequency dispersion and mode spilt in the arch. The responses of the arch with cracks are simulated under the broad-band, narrow-band and chirp excitations. In order to estimate the spatial, time and frequency concentrations of responses, the reciprocal length, time-frequency transform and correlation coefficient are introduced in this investigation.

Surrogate POD Models for Parametrized Sheet Metal Forming Applications

The aim of this work is to present a POD (Proper Orthogonal Decomposition) based surrogate approach for sheet metal forming parametrized applications. The final displacement field for the stamped work-piece computed using a finite element approach is approximated using the method of snapshots for POD mode determination and kriging for POD coefficients interpolation. An error analysis, performed using a validation set, shows that the accuracy of the surrogate POD model is excellent for the representation of finite element displacement fields. A possible use of the surrogate to assess the quality of the stamped sheet is considered. The Green-Lagrange strain tensor is derived and forming limit diagrams are computed on the fly for any point of the design space. Furthermore, the minimization of a cost function based on the surrogate POD model is performed showing its potential for solving optimization problems.