Linear Shape Functions Research Articles

Hybrid singular elements in fracture mechanics

A method using the concept of singular hybrid finite element for the determination of the stress intensity factor and the T stress of cracked panels submitted to plane stress and plane strain states is presented. A complete formulation of a hybrid equilibrium singular superelement is presented. This superelement includes linear, bilinear, and quadratic shape functions, allowing its use with linear, bilinear, and quadratic displacementbased conventional finite elements. The method is applied to various typical configurations of cracked rectangular plates, including stiffened panels. The results obtained are discussed and compared with previous available data.

A Finite Element Approach for the Simulation of Quasi‐Brittle Fracture

AbstractIn the context of a strong discontinuity approach, we propose a finite element formulation with an embedded displacement discontinuity. The basic assumption of the proposed approach is the additive split of the total displacement field in a continuous and a discontinuous part. An arbitrary crack splits the linear triangular finite element into two parts, namely a triangular and a quadrilateral part. The discontinuous part of the displacement field in the quadrilateral portion is approximated using linear shape functions. For these purposes, the quadrilateral portion is divided into two triangular parts which is in this way similar to the approach proposed in [5]. In contrast, the discretisation is different compared to formulations proposed in [1] and [3], where the discontinuous part of the displacement field is approximated using bilinear shape functions. The basic theory of the underlying finite element formulation and a cohesive interface model to simulate brittle fracture are presented. By means of representative numerical examples differences and similarities of the present formulation and the formulations proposed in [1] and [3] are highlighted. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A discontinuous finite element formulation for Helmholtz equation

A discontinuous finite element formulation for Helmholtz equation

Least-squares finite element method for the advection–diffusion equation

Least-squares finite element method for the advection–diffusion equation

Evaluation of the mechanical deformation in incompressible linear and nonlinear magnetic materials using various electromagnetic force density methods

Mechanical deformation in incompressible linear and nonlinear magnetic materials was evaluated using various conventional electromagnetic volume and surface force density methods. These conventional force density methods are the Maxwell stress tensor method, Korteweg-Helmholtz force density method (KH), magnetic charge method, magnetizing current method, and Kelvin force density method (KV). The total force values obtained using these different force density methods were found to be the same and equal to the total force using the principle of virtual work, but the distribution of force density values calculated using the given force density methods was found to be different from each other. Using the given five force density methods, the mechanical deformations were evaluated and compared to one another. The KH and KV in incompressible material were shown to give the same mechanical deformation by employing the finite element method (FEM), verifying the theoretical equivalence. To implement the KV, the derivative of magnetic field intensity with respect to the geometrical position was calculated using a linear shape function of FEM along with the nodal field values in each element. A magnetic systems was tested to compare the mechanical deformation in linear and nonlinear magnetic materials.

Piezo-control of forced vibrations of a thermoelastic composite plate

Piezo-control of forced vibrations of a thermoelastic composite plate

PYROELECTRIC EFFECT ON DYNAMIC RESPONSE OF COUPLED DISTRIBUTED PIEZOTHERMOELASTIC COMPOSITE PLATE

ABSTRACT The equations governing the linear response of a coupled piezothermoelastic composite plate, considering the pyroelectric effect, are outlined. The finite element method with linear shape functions for a first-order shear deformation theory of laminated plates is used to obtain the deflections and temperature. The finite element equations are developed for a piezothermoelectric sensor and actuator. The governing equations are solved using the time-marching method. Numerical results are presented for the conventional active direct proportional control of the plate. Controlled and uncontrolled displacements and temperature responses are graphically illustrated. The corresponding temperature response with consideration of the pyroelectric effects is also presented. Dynamic case studies are carried out to observe the pyroelectric effects in the piezothermoelastic control systems.

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Free vibration analysis of isotropic rectangular plates using a high-order triangular finite element with shear

Free vibration analysis of isotropic rectangular plates using a high-order triangular finite element with shear

A partition method for the solution of a coupled liquid-structure interaction problem

A partition method for the solution of a coupled liquid-structure interaction problem

DYNAMIC ANALYSIS OF DISTRIBUTED PIEZOTHERMOELASTIC COMPOSITE PLATE USING FIRST-ORDER SHEAR DEFORMATION THEORY

Equations governing the linear response of a piezothermoelastic plate based on Hamilton's principle and finite element methods with linear shape functions for a first-order shear deformation theory of laminated plates are outlined. The governing equations are solved using the time marching method. Numerical results are presented for cases of a prescribed thermal loading and the conventional active negative control piezoelectric analysis. Controlled and uncontrolled responses are compared graphically. The numerical studies demonstrate the importance of the thermal environment, as well as the effectiveness of active compensation of thermal deformations using piezoelectric structures. It is found that the displacements caused by temperature change are important in the precision sensing and control of distributed systems by piezoelectric materials.

Thermoelastohydrodynamic (TEHD) Analysis of a Grooved Thrust Washer

A numerical model is developed to analyze the influence of thermal deformations on the performance of a radially grooved thrust washer. The analysis couples the flow phenomena (including cavitation) in the lubricant with the heat transfer in both the fluid and solid media, as well as with the thermally induced deformations in the solid parts. The finite element method (both two-dimensional and three-dimensional, with linear and quadratic shape functions) is used to solve the Reynolds equation for flow, the energy equation for temperature and the thermo-elasticity equations for deformations in the solid. Grid coupling is achieved by using a Newton-Raphson iteration. Realistic boundary conditions and geometry are used for the fluid and solid domains. The results show that, for the case of a properly shaped stator, the thermal deformations can lead to an increase in bearing performance.

Virtual crack closure technique: History, approach, and applications

An overview of the virtual crack closure technique is presented. The approach used is discussed, the history summarized, and insight into its applications provided. Equations for two-dimensional quadrilateral finite elements with linear and quadratic shape functions are given. Formulas for applying the technique in conjunction with three-dimensional solid elements as well as plate/shell elements are also provided. Necessary modifications for the use of the method with geometrically nonlinear finite element analysis and corrections required for elements at the crack tip with different lengths and widths are discussed. The problems associated with cracks or delaminations propagating between different materials are mentioned briefly, as well as a strategy to minimize these problems. Due to an increased interest in using a fracture mechanics–based approach to assess the damage tolerance of composite structures in the design phase and during certification, the engineering problems selected as examples and given as references focus on the application of the technique to components made of composite materials.

HRR特異性を有する有限要素の検討 : 一次元特異要素(GS7 特異場・じん性評価)

In analyzing a cracked solid by using the finite element method, the singularity of the tip of crack must be taken into consideration and for this purpose the so-called singular element which has either Ο(γ^<-1/2>) singularity or Ο(γ^<-1>) singularity have been developed so far. The order of singularity of the tip of a crack in a strain-hardening solid is different from that in a linear-elastic solid and it becomes Ο(γ^<-1/2>)〜Ο(γ^<-1>) according to the work hardening exponent of the solid. This singularity is called as the HRR singularity. In our series of researches, the HRR singular elements will be examined from the one-dimensional case to the three-dimensional one by using the super-parametric elements. One-dimensional HRR singular elements will be focused in the present research. Consequently, the singular elements with Ο(γ^<-(l-1)/l>) singularity can be developed successfully by using the super-parametric element defined by the linear shape function and l-th order of the polynomial function defining the geometry and moving its mid-side nodes to the appropriate site on the edge of element.

HRR特異性を有する有限要素の検討 : 三次元特異要素(GS7 特異場・じん性評価)

In our previous research, the HRR singular elements were examined for the one-dimensional case by using the super-parametric elements. Three-dimensional HRR singular elements are focused in the present research. The super-parametric element defined by the linear shape function and l-th order of the polynomial function defining the geometry is used as well as one-dimensional case. The mid-side nodes of such a super-parametric element are moved to the appropriate sites on its edge and one of its face is collapsed to one line according to the method presented by Hussain et al. The resultant shape of the element becomes wedge. Consequently, we can verify that this wedge shape element has Ο(γ^<-(l-1)/l>) singularity.

Three-dimensional hierarchical finite element free vibration analysis of annular sector plates

Three-dimensional hierarchical finite element free vibration analysis of annular sector plates

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

AbstractA numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Experimental and numerical investigations of axisymmetric wave propagation in cylindrical pipe filled with fluid.

Acoustic wave propagation in fluid-filled cylindrical pipe with arbitrary thickness is investigated numerically and experimentally. The vibrational properties of the coupled fluid-pipe system are evaluated by a layerwise approach, which is similar to the finite-strip method. In this approach, the thick cylindrical wall is divided into a number of thin cylindrical layers in the thickness direction. The displacements in the thickness direction for each layer are approximated by linear-shape functions. The governing equation is obtained by using an energy minimization principle. The dispersion curves, distribution of vibrational energy between pipe wall and contained fluid, and displacement fields are examined. The dependence of the dispersion curves on wall thickness is discussed. Two PZT ring transducers adhered to the outer surface of pipe are used as source and receiver, respectively. The propagating waves generated by burst signals are measured. To localize transient signal both in time and frequency domains, the discrete wavelet transform is applied to decomposing the receiving signal into several components. Each component is limited to a narrower bandwidth. Therefore the frequency-dependent group velocity is estimated. The experimental and numerical results are compared.

비정렬 격자계를 이용한 희박기체 영역의 3차원 병렬 직접모사법 개발

본 연구에서는 비정렬 격자를 이용한 병렬화된 3차원 직접모사법을 개발하였다. 모사 입자의 추적에는 선형 형상 함수를 3차원으로 확장하여 적용하였다. 프로그램에서는 효율적인 병렬계산을 위하여 모사 입자의 수에 기초한 영역 재분할 방식을 이용하여 프로세서간의 작업 균형을 유지하였다. 밀도 차가 큰 두 개 이상의 유동이 동시에 존재할 경우를 효과적으로 모사하기 위해서 입자에 가중치를 적용하는 기법을 도입하였다. 프로그램의 검증을 위해서 3차원 삼각 날개 유동을 해석하여 실험치나 다른 계산 결과와 비교하였다. 또한 고도 100km의 희박한 환경헤서 우주발사체 탑재부 주의 유동을 해석하여 노즐 전방으로 고온의 풀룸이 역류하여 비행체 표면에 영향을 미칠 수 있는 가능성을 타진하였다. In the present study, a 3-D Parallel DSMC method in developed on unstructured meshes for the efficient simulation of rarefied gas flows. Particle tracing between cells in achieved based on a linear shape function extended to three dimensions. For high parallel efficiency, successive domain decomposition is applied to achieve load balancing between processors by accounting for the number of particles. A particle weighting technique is also adopted to handle flows containing gases of significantly dirrerent number densities in the same flow domain. Application is made for flow past a 3-D delta wing and the result is compared with that from experiment and other calculation. Flow around a rocket payload at 100km altitude is also solved and the effect of plume back flow from the nozzle in studied.

The boundary contour method for piezoelectric media with linear boundary elements

AbstractThis paper presents a development of the boundary contour method (BCM) for piezoelectric media. First, the divergence‐free property of the integrand of the piezoelectric boundary element is proved. Secondly, the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and Green's functions (Computer Methods in Applied Mechanics and Engineering 1998; 158: 65) for piezoelectric media. The BCM is applied to the problem of piezoelectric media. Finally, numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method (BEM). The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified. Copyright © 2003 John Wiley &amp; Sons, Ltd.