Special Shape Functions Research Articles

A new crack-tip element for the logarithmic stress-singularity of Mode-III cracks in spring interfaces

A new crack-tip finite element able to improve the accuracy of Finite Element Method (FEM) solutions for cracks growing along the Winkler-type spring interfaces between linear elastic adherents is proposed. The spring model for interface fracture, sometimes called Linear-Elastic (perfectly) Brittle Interface Model (LEBIM), can be used, e.g., to analyse fracture of adhesive joints with a thin adhesive layer. Recently an analytical expression for the asymptotic elastic solution with logarithmic stress-singularity at the interface crack tip considering spring-like interface behaviour under fracture Mode III was deduced by some of the authors. Based on this asymptotic solution, a special 5-node triangular crack-tip finite element is developed. The generated special singular shape functions reproduce the radial behaviour of the first main term and shadow terms of the asymptotic solution. This special element implemented in a FEM code written in Matlab has successfully passed various patch tests with spring boundary conditions. The new element allows to model cracks in spring interfaces without the need of using excessively refined FEM meshes, which is one of the current disadvantages in the use of LEBIM when stiff spring interfaces are considered. Numerical tests carried out by h-refinement of uniform meshes show that the new singular element consistently provides significantly more accurate results than the standard finite elements, especially for stiff interfaces, which could be relevant for practical applications minimizing computational costs. The new element can also be used to solve other problems with logarithmic stress-singularities.

Analysis of the sound insulation performance of periodic wall structures by a virtual acoustic laboratory

The propagation of acoustic waves in periodic structures, also known as phononic crystals or PCs, is prohibited in certain frequency ranges, which are referred to as the frequency band-gaps. The existence, the location and the width of the frequency band-gaps are mainly determined by the geometrical parameters and the material properties of the PCs. In this work, a virtual acoustic laboratory based on a fully coupled fluid-structure interaction (FSI) model is developed to determine the sound insulation capacity of one-dimensional (1D) and two-dimensional (2D) periodic walls. The FSI model is discretized using the frequency-domain spectral element method (FDSEM), which is an advanced finite element method (FEM) using special high-order shape functions. Following the guidelines of the ISO10140, the setup of the developed FSI model allows us to take into account the essential physical phenomena, especially the interaction of the wall structure with the fluid domains (air). The FSI model based on the FDSEM increases the computational efficiency and accuracy in comparison with the standard FEM. Several numerical examples will be presented and discussed to show that the designed periodic walls in certain frequency ranges within the band-gaps may exhibit much better sound insulation capabilities than monolithic walls.

An enriched finite element/level-set model for two-phase electrohydrodynamic simulations

In this work, a numerical model for the simulation of two-phase electrohydrodynamic (EHD) problems is proposed. It is characterized by a physically consistent treatment of surface tension as well as a jump in the electric material properties. The formulation is based on a finite element method enriched with special shape functions, capable of accurate capturing discontinuities both in the fluid pressure and the gradient of the electric potential. Phase interface is, thus, represented as a zero-thickness boundary. The proposed methodology allows modeling the electric force as an interfacial one, strictly abiding with the physics. The approach is tested using the droplet deformation benchmarks. Moreover, application of the method to study a three-dimensional (3D) case, not characterized by symmetry of revolution, is shown. The proposed methodology defines a basis for an enriched finite element method for a wide range of EHD problems.

A Finite Element Method for Modeling Edge Effects in Composite Laminates with Different Ply Orientations

A finite element (FE) method is used for modeling edge effects in composite laminates with different ply orientations. Two types of three-dimensional isoperimetric FE methods have been used. One in the interior region has standard linear shape functions, while the other in the exterior region has special shape functions for which an exponential variation is derived using the perturbation technique. The exponential shape functions are introduced to an originally written FE program which uses special mesh generation and numerical integration, as well as other subroutines. The proposed FE solution is found to be in good agreement with the solutions available in the literature. KEYWORDS: Finite-element modeling, Perturbation technique, Laminated composites, Edge effect, Delamination

Adaptive Exponential Finite Elements for a Phase Field Fracture Model

AbstractFracture phenomena can be described by a phase field model in which an independent scalar field variable in addition to the mechanical displacement is considered [3]. This field approximates crack surfaces as a continuous transition zone from a value that indicates intact material to another value that represents the crack. For an accurate approximation of cracks, narrow transition zones resulting in steep gradients of the fracture field are required. This necessitates a high mesh density in finite element simulations, which leads to an increased computational effort. In order to circumvent this problem without forfeiting accuracy, exponential shape functions were introduced in the discretization of the phase field variable, see [4]. These special shape functions allow for a better approximation of steep gradients of the phase field with less elements as compared to standard Lagrange elements. Unfortunately, the orientation of the exponential shape functions is not uniquely determined and needs to be set up in the correct way in order to improve the approximation of smooth cracks. This work solves the issue by adaptively reorientating the exponential shape functions according to the nodal values of the phase field gradient in each element. Furthermore, a local approach is pursued that uses exponential shape function only in the vicinity of the crack, whereas standard bilinear shape function are used away from the crack.

Higher Order Multiscale Finite Element Method for Heat Transfer Modeling.

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

Adaptive numerical integration of exponential finite elements for a phase field fracture model

Phase field models for fracture are energy-based and employ a continuous field variable, the phase field, to indicate cracks. The width of the transition zone of this field variable between damaged and intact regions is controlled by a regularization parameter. Narrow transition zones are required for a good approximation of the fracture energy which involves steep gradients of the phase field. This demands a high mesh density in finite element simulations if 4-node elements with standard bilinear shape functions are used. In order to improve the quality of the results with coarser meshes, exponential shape functions derived from the analytic solution of the 1D model are introduced for the discretization of the phase field variable. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the fracture field. Unfortunately, lower-order Gauss-Legendre quadrature schemes, which are sufficiently accurate for the integration of bilinear shape functions, are not sufficient for an accurate integration of the exponential shape functions. Therefore in this work, the numerical accuracy of higher-order Gauss-Legendre formulas and a double exponential formula for numerical integration is analyzed.

Adaptive Orientation of Exponential Finite Elements for a Phase Field Fracture Model

AbstractOne technique to describe the failure of mechanical structures is a phase field model for fracture. Phase field models for fracture consider an independent scalar field variable in addition to the mechanical displacement [1]. The phase field ansatz approximates crack surfaces as a continuous transition zone in which the phase field variable varies from a value that indicates intact material to another value that represents cracks. For a good approximation of cracks, these transition zones are required to be narrow, which leads to steep gradients in the fracture field. As a consequence, the required mesh density in a finite element simulation and thus the computational effort increases. In order to circumvent this efficiency problem, exponential shape functions were introduced in the discretization of the phase field variable, see [2]. Compared to the bilinear shape functions these special shape functions allow for a better approximation of the steep transition with less elements. Unfortunately, the exponential shape functions are not symmetric, which requires a certain orientation of elements relative to the crack surfaces. This adaptation is not uniquely determined and needs to be set up in the correct way in order to improve the approximation of smooth cracks. The issue is solved in this work by reorientating the exponential shape functions according to the nodal value of phase field gradient in a particular element. To be precise, this work discusses an adaptive algorithm that implements such a reorientation for 2d and 3d situations.

Towards asphalt concrete modeling by the multiscale finite element method

Reliable numerical modeling of asphalt concrete (AC) is a complex problem due to a non periodic random structure of this material and a nonlinear behavior of its interacting constituents. Phenomena observed at the lower resolution highly influence the overall response of pavement layers made of asphalt concrete. In this paper, we focus on the selected aspects of its efficient numerical modeling using the viscoelastic Burgers material model. In this paper, we focus on two important components of the modeling process. Firstly, we present a straightforward algorithm for the random AC structure generation, which is automatically linked with the finite element (FE) mesh generation. It is based on the predefined shapes of inclusions and the relocation of nodes of the regular grid. Such an approach provides definitely coarser (but related to the microstructure) FE meshes than other ones used for this purposes. Thus, it saves a lot of computer resources at this level of the modeling process. Our next development concerns further reduction of the number of degrees of freedom (NDOF) that are necessary to tackle the heterogeneous domain. We take advantage of the multiscale finite element method (MsFEM) to solve given problems using a coarse macroscale mesh. Within its elements, we construct special shape functions of order 1, 2 or 3 revealing the complexity of the microresolution. We propose a novel extension of the linear elastic analysis to the viscoelastic one in terms of the MsFEM. Combining the aforementioned approaches, we provide a versatile tool for the numerical modeling of asphalt concrete. The proposed framework is illustrated with the examples of the special shape function constructions and the solutions to the compression tests of the samples made of AC. Obtained results confirm the efficiency and applicability of the proposed methodology.

Decoupled scaled boundary finite element method for analysing dam–reservoir dynamic interaction

In this study, an efficient method is developed for solving systems of partial differential equations governing seismic analysis of 2D dam reservoir interaction, in the frequency domain. Using Chebyshev higher-order polynomials as mapping function, special shape function, integration method of Clenshaw–Curtis and the integral form used to weighted residual method, coefficient matrices of the system of governing equations become diagonal. This means that the governing partial differential equation for each degree of freedom becomes independent from others. This feature and discretizing only boundaries of domain significantly reduce computational costs in comparison with other methods. In this regard, various problems such as dynamic analysis of empty gravity dam, calculating the hydrodynamic pressure on rigid dam, and dam–reservoir interaction analysis due to the horizontal motion of foundation are examined. Comparing the results of this method with other analytical/numerical methods shows high capability and accuracy of the proposed method.

Selected series method on buckling design of stiffened steel-concrete composite plates

The steel-concrete (SC) or steel-concrete-steel (SCS) composite structures are widely used in large-scale structures like immersed tunnels, protection structures, etc., due to their superior performances in capacity, ductility, waterproofness, construction efficiency, etc. The steel plates in these structures are usually stiffened with ribs to increase the out-of-plane rigidity and to improve the local buckling performance. While there have been several applications, the buckling design of the composite stiffened plates, based on the methods for steel structures, is still immature and conservative. To deal with this problem, a selected series method (SSM) based the energy theory is proposed, which could both be utilized in composite stiffened plates and steel stiffened plates. The special series shape functions, which could efficiently and accurately represent the 3-dimentional spatial buckling deformation of the stiffened plates with multiple stiffeners, are selected. The explicit high-order solutions for the buckling stress of stiffened plates are derived. Furthermore, a finite element model (FEM) is established. Based on the FEM, a numerical database including most common design parameters is derived, and extensive numerical analyses are conducted to modify and verify the proposed SSM. It is found that, for steel structures, the SSM has a 10%–20% improvement in accuracy compared with the existing methods, and for composite structures, the SSM fills the blank in this area and has acceptable accuracy for design. Finally, from the above studies, the design suggestions are given and discussed.

Boundary element method for fracture mechanics analysis of 3d non-planar cracks in anisotropic solids

Among the numerical approaches to fracture mechanics analysis of cracked anisotropic solids, the boundary element method is notable for high accuracy and performance due to its semi-analytical nature and the use of only boundary mesh. Various boundary element techniques were proposed for 3D fracture mechanics analysis. However, the main problem of these approaches is the treatment of singular and hypersingular integrals, which can demand analytic evaluation of coefficient at kernel singularity at singular point in local curvilinear coordinates, which produce cumbersome equations in the case of non-planar geometries. Therefore, the paper presents novel formulation of the boundary element method for 3D fracture mechanics analysis of anisotropic solids with non-planar cracks. Pan’s single domain boundary element formulation is extended with several novelties, which allow accurate analysis of non-planar geometries. These are modified Kutt’s numerical quadratures with Chebyshev nodes for accurate evaluation of singular and hypersingular integrals; polynomial mappings for smoothing the integrand at the crack front line; and special shape functions, which account for a square-root stress singularity at crack front and allow accurate determination of stress intensity factors. The kernels of boundary integral equations are evaluated using the exponentially convergent quadrature, which allows derivation of fast boundary element technique. The procedure for accurate numerical determination of stress intensity factors at arbitrary point of the crack front is also developed. Numerical examples are presented, which show high accuracy of the proposed boundary element method. It is shown that non-planar cracks exhibit shearing mode opening along with normal one due to its geometry and direction of the applied loading. Present approach can be combined with Sih’s strain energy density criterion to study 3D cracks propagation in anisotropic elastic solids under fatigue loading, which is the direction of future research.

Asymptotically flat wormhole solutions with variable equation-of-state parameter

This paper discusses exact wormhole solutions in the framework of general relativity with a general equation of state that reduced to a linear equation of state asymptotically. By considering a special shape function, we find classes of solutions which are asymptotically flat. We study the violation of the null energy condition as the main ingredient in wormhole physics. The possibility of finding wormhole solutions with asymptotically different state parameter is investigated. We show that in principle, a wormhole with a vanishing redshift function and the selected shape function, cannot satisfy the null energy condition at a large distance from the wormhole. Our solutions have the positive total amount of matter in the ``volume integral quantifier'' method. For this class of solutions, fluid near the wormhole throat is in the phantom regime which at some $r={r}_{2}$, the phantom regime is connected to a distribution of dark energy regime. Thus, we need a small amount of exotic matter to construct wormhole solutions.

Boundary element analysis of 3D shell‐like rigid electrically conducting inclusions in anisotropic thermomagnetoelectroelastic solids

AbstractThe paper presents general boundary element approach for analysis of thermomagnetoelectroelastic solids containing shell‐like electrically conducting inclusions with high magnetic permittivity. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric and magnetic potentials of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell‐like inclusion. Special shape functions are introduced, which account for this square‐root singularity and allow accurate determination of field intensity factors. Approaches for fast and accurate numerical evaluation of anisotropic thermomagnetoelectroelastic kernels are discussed, which are crucial in derivation of fast and precise boundary element approach. Numerical examples are presented.

A polytree based coupling method for non-matching meshes in 3D

The coupling of non-matching meshes is of great interest in the field of computational mechanics. It reduces meshing burden by allowing users to divide a problem domain into several simpler subdomains and generate meshes for them independently. In this paper, a coupling method using arbitrary polyhedron elements is presented. On the interface of two non-matching meshes, a new surface mesh of polygon elements is constructed by merging the non-matching meshes. A shifting procedure is designed to prevent the distortion of surface elements. The polyhedron elements connected to the interface are modified by replacing their faces on the interface with the new polygon elements, leading to matching discretization on the interface. The modified polyhedron elements are modeled by the scaled boundary finite element method and no further volume discretization is required. No interface constraints or special shape functions are required in the analysis. Seven numerical examples are presented to demonstrate the versatility and robustness of the proposed procedure. Convergence behavior is also examined.

Boundary Element Modeling of Pyroelectric Solids with Shell Inclusions

Abstract The paper presents general boundary element approach for analysis of thermoelectroelastic (pyroelectric) solids containing shell-like electricity conducting permittive inclusions. The latter are modeled with opened surfaces with certain boundary conditions on their faces. Rigid displacement and rotation, along with constant electric potential of inclusions are accounted for in these boundary conditions. Formulated boundary value problem is reduced to a system of singular boundary integral equations, which is solved numerically by the boundary element method. Special attention is paid to the field singularity at the front line of a shell-like inclusion. Special shape functions are introduced, which account for this square-root singularity and allow accurate determination of field intensity factors. Numerical examples are presented.

Construction of special shape functions for triangular elements with one edge lying in the crack front

This paper presents a new and general construction method of special triangular elements with one edge lying in the crack front. In our method, principal items which represent the asymptotic properties around the crack front in local coordinate system are constructed. Based on the principal items, two groups of basis functions for crack front elements are introduced. Combining undetermined coefficients and basis functions, two types of crack front elements, which are denoted by low order crack front elements and high order ones, are constructed. Secondly, a displacement extrapolation technique with these newly developed elements is proposed. Applying this technique in the BEM applications for crack problems, numerical difficulties, which arise when two vertexes of the assisted line both locate inside the element, can be circumvented conveniently. Finally, two types of elements are employed in the dual boundary element codes for crack problems. Numerical examples are presented to verify the three types of elements. Results show that the stress intensity factors computed by employing these three developed elements are all very accurate compared with the existing results.

Boundary Element Analysis of Anisotropic Thermomagnetoelectroelastic Solids with 3D Shell-Like Inclusions

Abstract The paper presents novel boundary element technique for analysis of anisotropic thermomagnetoelectroelastic solids containing cracks and thin shell-like soft inclusions. Dual boundary integral equations of heat conduction and thermomagnetoelectroelasticity are derived, which do not contain volume integrals in the absence of distributed body heat and extended body forces. Models of 3D soft thermomagnetoelectroelastic thin inclusions are adopted. The issues on the boundary element solution of obtained equations are discussed. The efficient techniques for numerical evaluation of kernels and singular and hypersingular integrals are discussed. Nonlin-ear polynomial mappings are adopted for smoothing the integrand at the inclusion’s front, which is advantageous for accurate evaluation of field intensity factors. Special shape functions are introduced, which account for a square-root singularity of extended stress and heat flux at the inclusion’s front. Numerical example is presented.

Edge Meshless Method Applied to Vector Electromagnetic Problems

A challenge in meshless methods dealing with vector electromagnetic problems is to produce numerical solutions that are free of spurious modes given that the generated vector field does not satisfy the condition of zero divergence. The edge meshless method (EMM) constructs its approximations using special shape functions based on edges to produce vector fields that are divergence free and to guarantee the continuity of the tangential field components. This paper presents the application of the EMM to solve vector electromagnetic problems. The 2-D Maxwell eigenvalue problem with anisotropic medium is tested to demonstrate that the technique produces correct numerical solution without spurious modes.

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

SummaryLinear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.