Set Of Shape Functions Research Articles

Some specific wormhole solutions in extended f(R,G,T) gravity

This research work provides an exhaustive investigation of the viability of different coupled wormhole (WH) geometries with the relativistic matter configurations in the [Formula: see text] extended gravity framework. We consider a specific model in the context of [Formula: see text]-gravity for this purpose. Also, we assume a static spherically symmetric spacetime geometry and a unique distribution of matter with a set of shape functions ([Formula: see text]) for analyzing different energy conditions. In addition to this, we examined WH-models in the equilibrium scenario by employing anisotropic fluid. The corresponding results are obtained using numerical methods and then presented using different plots. In this case, [Formula: see text] gravity generates additional curvature quantities, which can be thought of as gravitational objects that maintain irregular WH-situations. Based on our findings, we conclude that in the absence of exotic matter, WH can exist in some specific regions of the parametric space using modified gravity model as [Formula: see text].

Construction of generalized shape functions over arbitrary polytopes based on scaled boundary finite element method's solution of Poisson's equation

SummaryA general technique to develop arbitrary‐sided polygonal elements based on the scaled boundary finite element method is presented. Shape functions are derived from the solution of the Poisson's equation in contrast to the well‐known Laplace shape functions that are only linearly complete. The application of the Poisson shape functions can be complete up to any specific order. The shape functions retain the advantage of the scaled boundary finite element method allowing direct formulation on polygons with arbitrary number of sides and quadtree meshes. The resulting formulation is similar to the finite element method where each field variable is interpolated by the same set of shape functions in parametric space and differs only in the integration of the stiffness and mass matrices. Well‐established finite element procedures can be applied with the developed shape functions, to solve a variety of engineering problems including, for example, coupled field problems, phase field fracture, and addressing volumetric locking in the near‐incompressibility limit by adopting a mixed formulation. Application of the formulation is demonstrated in several engineering problems. Optimal convergence rates are observed.

Extension of the unsymmetric 8‐node hexahedral solid element US‐ATFH8 to 3D hyper‐elastic finite deformation analysis

AbstractThis work extends the recent US‐ATFH8 element to 3D hyper‐elastic finite deformation analysis. Using two sets of shape functions, the new 3D element comprises of 8 nodes and 24 DOFs. The first set of shape functions represent the test functions that come from the conventional isoparametric interpolation, and the second set, representing the trial functions, are constructed from the homogenous solutions for linear elasticity governing equations, termed analytical trial functions (ATFs). This study considers finite deformation for hyper‐elastic materials, but it is assumed that the analytical solutions associated with hyper‐elastic materials can be updated to hold approximately in each incremental step. Moreover, the deformation information required for stress computation is updated by using the incremental deformation gradient, which is constructed from the updated ATFs. Numerical examples show that without additional pressure DOF, the element US‐ATFH8 still behaves well in nearly incompressible hyper‐elastic 3D problems with finite deformation, even when the meshes are extremely distorted.

A duality‐based coupling of Cosserat crystal plasticity and phase field theories for modeling grain refinement

AbstractHigh‐rate deformation processes of metals entail intense grain refinement and special attention needs to be paid to capture the evolution of microstructure. In this article, a new formulation for coupling Cosserat crystal plasticity and phase field is developed. A common approach is to penalize kinematic incompatibility between lattice orientation and displacement‐based elastic rotation. However, this can lead to significant solution sensitivity to the penalty parameter, resulting in low accuracy and convergence rates. To address these issues, a duality‐based formulation is developed which directly imposes the rotational kinematic compatibility. A weak inf‐sup‐based skew‐symmetric stress projection is introduced to suppress instabilities present in the dual formulation. An additional least squares stabilization is introduced to suppress the spurious lattice rotation with a suitable parameter range derived analytically and validated numerically. The required high‐order continuity is attained by the reproducing kernel approximation. It is observed that equal order displacement‐rotation‐phase field approximations are stable, which allows efficient employment of the same set of shape functions for all independent variables. The proposed formulation is shown to yield superior accuracy and convergence with marginal parameter sensitivity compared to the penalty‐based approach and successfully captures the dominant rotational recrystallization mechanism including block dislocation structures and grain boundary migration.

A novel [formula omitted] continuity finite element based on Mindlin theory for doubly-curved laminated composite shells

As a widely used structural form, doubly-curved composite shells have been applied in aviation and other engineering fields. With a refined mechanical model, structural performances can be accurately predicted to help designers choosing best geometrical and material parameters. This work introduces a novel rectangular finite element for doubly-curved laminated composite shells based on a new set of strain-based shape functions. The governing equations are established on the Mindlin shell theory (a type of first order shear deformable shell theory), which incorporates a rectification of shear correction factor for laminates and von Kármán type nonlinearity. Based on strain approach, the shape functions for rectangular element are assumed as polynomial of 28 parameters to consider the influence of shear effect. Apart from 20 geometrical relations of four element nodes, shear force equilibrium equations are introduced to offer the additional eight equations to derive a new set of shape functions for finite element model. Using shape functions, a novel rectangular shell element is proposed for doubly curved laminated shell, which also maintains a compatibility with Kirchhoff shells. Numerical results for linear static bending, dynamic vibration and nonlinear bending cases of flat plate, cylindrical shell and doubly-curved laminated shell are compared with the available results in literatures and ABAQUS simulation for the sake of validating the present method.

Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

In this contribution, the variational problem for the Kirchhoff plate based on the modified strain gradient theory (MSGT) is derived, and the Euler-Lagrange equations governing the equation of motion are obtained. The Galerkin-type weak form, upon which the finite element method is constructed, is derived from the variational problem. The shape functions which satisfy the governing homogeneous partial differential equation are derived as extensions of Adini-Clough-Melosh (ACM) and Bogner-Fox-Schmit (BFS) plate element formulations by introducing additional curvature degrees of freedom (DOF) on each node. Based on the proposed set of shape functions, 20-, 24-, 28- and 32- DOF modified strain gradient theory-based higher-order Kirchhoff microplate element are proposed. The performance of the elements are demonstrated in terms of various tests and representative boundary value problems. Length scale parameters for gold are also proposed based on experiments reported in literature.

A polynomial based dynamic expansion and data consistency assessment and modification for cylindrical shell structures

A designated set of shape functions are created for cylindrical shell structures. The 1D Chebyshev polynomial is applied along the axial direction and the 1D harmonic basis along the circumferential direction. A 2D orthogonal space is formed on the cylindrical domain by combining the above two 1D bases and is used to develop the expansion function. A finite element model is not required for the expansion technique, nor are the boundary conditions or external forcing functions. Using these generated shape functions, the dynamic time response at some points on cylindrical shells structure can be expanded to obtain the information at a larger number of points. Only the measurement data, geometry, and coordinates of measured points are needed. The developed polynomial expansion technique is also extended to a Data Consistency Assessment Function (DCAF), which can assess the consistency of the data set and identify the inconsistent points. The inconsistent points can also be modified by using the polynomial expansion method. A cylindrical shell structure in either air or water medium is used to experimentally demonstrate the use of the developed shape functions for polynomial expansion and data consistency assessment and modification.

Efficient modelling of beam-like structures with general non-prismatic, curved geometry

The analysis of three-dimensional (3D) stress states can be complex and computationally expensive, especially when large deflections cause a nonlinear structural response. Slender structures are conventionally modelled as one-dimensional beams but even these rather simpler analyses can become complicated, e.g. for variable cross-sections and planforms (i.e. non-prismatic curved beams). In this paper, we present an alternative procedure based on the recently developed Unified Formulation in which the kinematic description of a beam builds upon two shape functions, one for the beam’s axis, the other for its cross-section. This approach predicts 3D displacement and stress fields accurately and is computationally efficient in comparison with 3D finite elements. However, current modelling capabilities are limited to the use of prismatic elements. As a means for further applicability, we propose a method to create beam elements with variable planform and variable cross-section, i.e. of general shape. This method employs an additional set of shape functions which describes the geometry of the structure exactly. These functions are different from those used for describing the kinematics and provide local curvilinear basis vectors upon which 3D Jacobian transformation matrices are produced to define non-prismatic elements. The model proposed is benchmarked against 3D finite element analyses, as well as analytical and experimental results available in the literature. Significant computational efficiency gains over 3D finite elements are observed for similar levels of accuracy, for both linear and geometrically nonlinear analyses.

Reduced-order models for the analysis of a vertical rod under parametric excitation

This paper focuses on the analysis of the parametric excitation of a vertical and immersed flexible rod, showing the influence of the choice of the shape function used in the Galerkin’s method. For this, three different reduced-order models (ROMs) are obtained from the continuous equation of transverse motion employing different shape functions. The first model (ROM(i)) uses an approximation of the actual vibration mode of the rod, written as a “Bessel-like” function. The second model (ROM(ii)) is based on a single trigonometric function as the shape function. Finally, a multi-modal ROM (ROM(iii)) is obtained using three trigonometric functions as a set of shape functions. Simulations are carried out aiming at verifying the capability of each model to properly represent the dynamics of the rod under parametric excitation. The quality of the numerical results obtained from the integration of the aforementioned ROMs is assessed by means of a comparison with a solution based on the finite element method (FEM). In addition to the numerical analysis, an analytical solution for the steady-state amplitude of a generic Duffing-Mathieu-Morrison oscillator is obtained using the method of multiple scales for the one degree of freedom ROMs. A case study is developed using the data of a vertical riser as an example of an engineering application. Maps of the steady-state amplitude as a function of the excitation amplitude and frequency are plotted using both the numerical simulations and the multiple scales solution. The results show that ROM(i) and ROM(iii) are in good agreement with the finite element solution. ROM(i) has the advantage of having only one degree of freedom and, consequently, can be studied using the analytical solution aforementioned. The use of a ROM with one degree of freedom using “Bessel-like” functions in the Galerkin’s scheme is concluded to have clear advantages from the practical point of view. The analytical solution allows this kind of ROM to give a post-critical amplitude map with low computational effort and that is in good agreement with the maps obtained with the simulation of the ROMs.

Non-Model Based Expansion from Limited Points to an Augmented Set of Points Using Chebyshev Polynomials

Expansion of measured dynamic response at a limited set of points is of great interest in the study of structural dynamic systems. Measurements at limited points on a structure can be expanded to unmeasured points by using System Equivalent Reduction Expansion Process (SEREP). However, a finite element model is required to obtain the mode shapes of the system. In this work, a non-model based expansion technique is proposed to expand the vibration characteristics at a sparse set of points to a much larger set of points without the use of a finite element model. Shape functions based on orthogonal polynomials can be used to decompose the deformation of vibrating structures into the summation of a set of shape functions with corresponding weighting coefficients. To obtain the vibration characteristics at an augmented set of points, measurements at a sparse set of points can be related to a set of shape functions which have the same mesh resolution as the sparse configuration. The vibration characteristics at an augmented set of points can then be reconstructed from the shape functions of the high resolution and scaled shape function weighting coefficients of the sparse configuration. Only a sparse set of measurements, geometry and coordinates of measured points are needed. In the work studied here, an analytical plate and a Base-Upright (BU) structure are used as samples to study the application of the proposed method. Both analysis and experiment are used to prove the concept.

A multiscale approach to hybrid RANS/LES wall modeling within a high‐order discontinuous Galerkin scheme using function enrichment

SummaryWe present a novel approach to hybrid Reynolds‐averaged Navier‐Stokes (RANS)/ large eddy simulation (LES) wall modeling based on function enrichment, which overcomes the common problem of the RANS‐LES transition and enables coarse meshes near the boundary. While the concept of function enrichment as an efficient discretization technique for turbulent boundary layers has been proposed in an earlier article by Krank & Wall (A new approach to wall modeling in LES of incompressible flow via function enrichment. J Comput Phys. 2016;316:94‐116), the contribution of this work is a rigorous derivation of a new multiscale turbulence modeling approach and a corresponding discontinuous Galerkin discretization scheme. In the near‐wall area, the Navier‐Stokes equations are explicitly solved for an LES and a RANS component in one single equation. This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard high‐order polynomial basis resolves turbulent eddies, where the mesh is sufficiently fine and the enrichment automatically computes the ensemble‐averaged flow if the LES mesh is too coarse. As a result of the derivation, the RANS model is applied solely to the RANS degrees of freedom, which effectively prevents the typical issue of a log‐layer mismatch in attached boundary layers. As the full Navier‐Stokes equations are solved in the boundary layer, spatial refinement gradually yields wall‐resolved LES with exact boundary conditions. Numerical tests show the outstanding characteristics of the wall model regarding grid independence, superiority compared to equilibrium wall models in separated flows, and achieve a speed‐up by two orders of magnitude compared to wall‐resolved LES.

Efficient aeroelastic beam modelling and the selection of a structural shape basis

This paper considers the selection of structural shape bases for the modelling of flexible beams using the nonlinear beam shapes formulation. The approach provides a low order geometrically nonlinear representation of a flexible beam using a kinematic description that is geometrically exact; a minimal state representation is achieved by projecting the problem onto a set of shape functions spanning the beam. Motivated by the requirements for the low-order modelling of aeroelastic systems, this study examines the practical requirements of applying these shape sets to a flexible wing problem. A number of candidate shape sets are presented: orthogonal and non-orthogonal, with and without load dependent boundary conditions. The efficiency with which this problem is solved is used to assess the performance of each candidate shape set and infer the common properties between the most efficient bases. For a system with a moderate number of states a fairly consistent performance is observed across all of the sets. For larger sized problems, orthogonality of the shape basis is required to ensure robustness and scalability. It is shown that for a generic basis set the necessary boundary conditions of the problem can be easily satisfied, furthermore any orthogonal properties of a chosen basis can be retained. This concept is extended to allow for the treatment of more complex configurations starting from a simple shape basis.

Frequency Analysis of Wing–Rotor System Considering Flexibility in Capsule Based on High-Accurate Method

Based on a typical wing–rotor thrust model on the airship, the dynamic influence of the gyroscopic effects from the tip rotor acting on the overall coupled system has been analyzed. Meanwhile, the flexibility at the capsule boundary has been studied, as well. Hamilton’s principle is employed to derive the general governing equations and the numerical Rayleigh–Ritz method is finally chosen in actual frequency computations. A new set of shape functions are put forward and verified which take most of the couplings among dimensions into account. The parameter studies are also conducted to make deep investigations. The results demonstrate that the inherent frequencies are significantly affected by the rotor speed and the flexible capsule condition. When rotor revolves, the modal shapes have reached into complex states and the components of each mode will change with the increment of rotor speed. The flexibility will also greatly reduce the entire frequencies compared with the rigid case. It is also demonstrated that the inherent property will be significantly affected by the mounting geometry, rotor inertia, the structural stiffness, and rotor speed.

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used.

Adaptive Isogeometric Digital Height Correlation: Application to Stretchable Electronics

AbstractA novel adaptive isogeometric digital height correlation (DHC) technique has been developed in which the set of shape functions, needed for discretization of the ill‐posed DHC problem, is autonomously optimized for each specific set of profilometric height images, without a priori knowledge of the kinematics of the experiment. To this end, an adaptive refinement scheme is implemented, which refines the shape functions in a hierarchical manner. This technique ensures local refinement, only in the areas where needed, which is beneficial for the noise robustness of the DHC problem. The main advantage of the method is that it can be applied in experiments where the deformation mechanisms are unknown in advance, thereby complicating the choice of suitable shape functions. The method is applied to a virtual experiment in order to provide a proof of concept. A second virtual experiment is executed with stretchable electronics interconnects, which entail localized buckles upon deformation with complex kinematics. In both cases, accurate results were obtained, demonstrating the beneficial aspects of the proposed method. Moreover, the technique performance on profilometric images of a real experiment with stretchable interconnects was demonstrated.

Novel approach to FE solution of crack problems in the Laplacian-based gradient elasticity

Stress/strain concentration around a crack tip in ceramic foam-like structures, where the characteristic size of the foam cell is comparable with a typical length over which field quantities change significantly, is analysed. It is convenient to replace the foam structure by an effective continuum which retains all necessary characteristic features of the foam. To this end a homogenized cracked open cell ceramic foam is analysed using the strain/stress model based upon an implicit dependence of the non-local stress and strain on the local stress and strain in the form of an inhomogeneous Helmholtz equation which is solved together with the equilibrium equations. To preserve the advantage of the uncoupled system of equations, which means that one set of equations is solved prior the other one and that two sets of shape functions can be chosen independently, a recurrent sequence of well posed boundary value problems is constructed which can be solved using a classical 8-nodes isoparametric element for 2-D problems. The key idea behind the suggested procedure consists in replacing the scale parameter l by a parameter increment Δl chosen arbitrarily small. The given small disturbance causes a redistribution of the local stress/displacement field. After the new local stress/displacement field is known, another disturbance is introduced and so on. The procedure is stopped when the solution, exhibiting boundary layer behaviour, leads to the cohesive-like zone with length reaching a value uniquely related to the scale parameter l.

A hybrid wave-mode formulation for the vibro-acoustic analysis of 2D periodic structures

In the framework of vibrational analysis of 2D periodic waveguides, Floquet–Bloch theorem is widely applied for the determination of wave dispersion characteristics. In this context, the Wave Finite Element Method (WFEM) combines Periodic Structure Theory (PST) with standard FE packages, enabling wave dispersion analysis of waveguides involving structurally realistic unit-cells. For such applications, the computational efficiency of the WFEM depends on the choice of the formulation and can lead to numerical issues, worsen by extensive computational cost. This paper presents a coupled wave-mode approach for the determination of wave dispersion characteristics in structurally advanced periodic structures. It combines two scales of model order reduction. At the unit-cell׳s scale, Component Mode Synthesis (CMS) provides the displacement field associated with local resonances of the periodic structure, while the free wave propagation is considered using a spectral problem projection on a reduced set of shape functions associated with propagating waves, thus providing considerable reduction of the computational cost. An application is provided for a bi-directionally stiffened panel and the influence of reduction parameters is discussed, as well as the robustness of the numerical results.

An Efficient Solution for Volume Integral Equation Based on Meshfree Scheme

An efficient solution for volume integral equation based on meshfree scheme is proposed. The volume current is represented by a set of shape functions in the method of moments (MoM), where the shape functions are constructed on some distributed nodes without explicitly requiring continuity conditions, so that the basis function in this letter is not reliant on the tessellation and can be used with non-conformal meshes. Numerical results demonstrate that scattering from objects can be computed efficiently with the multilevel fast multi-pole method (MLFMM).

An analytical approach for the cross-sectional analysis of generalised beam theory

This paper presents an analytical approach for the cross-sectional analysis carried out within the framework of the generalised beam theory. With this method, the in-plane deformations of the conventional modes are determined as the dynamic modes of an unconstrained planar frame, from which the corresponding warping distributions are post-processed. The shape functions adopted for the transverse displacements of the stiffness elements used for the dynamic analysis are derived based on the analytical solutions of the beam problem. In the case of the axial displacements, two sets of shape functions are considered, which correspond to a linear polynomial representation as well as to the analytical dynamic solution. The differences between these two approximations are discussed in relation to the condition of inextensibility applied within the generalised beam theory to each plate segment forming the cross-section. With the proposed procedure, the conventional modes, which include the distortional, shear and local modes, do not need to be calculated separately, as carried out with the classical generalised beam theory, and the approach is applicable to open, partially closed and closed sections. Finally, the accuracy of the proposed approach is outlined by means of two numerical case studies.

A reduced formulation for the free-wave propagation analysis in composite structures

This paper presents an improved numerical strategy for the broadband analysis of wave propagation in composite or complex cross-sectional waveguides using the wave finite element method (WFE). Numerical analysis of such structures require highly discretized finite element models and leads to extensive computations. The proposed formulation relies on a projection of the cross-sectional transfer matrices on a reduced set of shape functions associated to propagating waves. Dispersion curves are then predicted only using a reduced number of eigenvectors. The performances and stability of this method are evaluated using the wavenumbers and wave shapes. Validations are provided for a sandwich composite beam and a cylindrical elasto-acoustic waveguide.