Hybrid Finite Element Formulation Research Articles

Fundamental-Solution-Based Hybrid Element Model for Nonlinear Heat Conduction Problems with Temperature-Dependent Material Properties

The boundary-type hybrid finite element formulation coupling the Kirchhoff transformation is proposed for the two-dimensional nonlinear heat conduction problems in solids with or without circular holes, and the thermal conductivity of material is assumed to be in terms of temperature change. The Kirchhoff transformation is firstly used to convert the nonlinear partial differential governing equation into a linear one by introducing the Kirchhoff variable, and then the new linear system is solved by the present hybrid finite element model, in which the proper fundamental solutions associated with some field points are used to approximate the element interior fields and the conventional shape functions are employed to approximate the element frame fields. The weak integral functional is developed to link these two fields and establish the stiffness equation with sparse and symmetric coefficient matrix. Finally, the algorithm is verified on several examples involving various expressions of thermal conductivity and existence of circular hole, and numerical results show good accuracy and stability.

A FUNDAMENTAL SOLUTION-BASED FINITE ELEMENT MODEL FOR ANALYZING MULTI-LAYER SKIN BURN INJURY

To understand the physiology of tissue burns for successful clinical treatment, it is important to investigate the thermal behavior of human skin tissue subjected to heat injury. In this paper, a fundamental solution-based hybrid finite element formulation is proposed for numerically simulating steady-state temperature distribution inside a multilayer human skin tissue during burning. In the present approach, since only element boundary integrals are involved, the computational dimension is reduced by one as the fundamental solutions used analytically satisfies the bioheat governing equation. Further, in multi-layer skin modeling, the burn is applied via a heating disk at constant temperature on a part of the epidermal surface of the skin tissue. Numerical results from the proposed approach are firstly verified by comparing them with exact solutions of a simple single-layered model or the results from conventional finite element method. Thereafter, a sensitivity analysis is carried out to reveal the effect of biological and environmental parameters on temperature distribution inside the skin tissue subjected to heat injury.

Effect of the electric conductivity on the modeling of the poling process of ferroelectric components

Piezoceramic materials, despite being semiconductors with a high but finite ohmic resistance, are generally modeled as perfect insulators. In this work the influence of the electric conductivity on the poling process of piezoceramic materials is investigated. To solve the electromechanically coupled boundary value problem, a reduced form of the Maxwell equations is implemented inside a hybrid finite element formulation. In this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of the remanent polarization. The material model is fully ferroelectric/ferroelastic coupled, whereas the material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. The simulation of poling processes for three different components are presented. First and second, the poling of a radially poled hollow cylinder and of a stack actuator is investigated. Here, a residual electric field appears after poling, leading to significant time-dependent changes of stresses and deformation due to subsequent charge transportation processes. The third example is a bending actuator composed of two layers of hard-PZT and soft-PZT material. It is shown that considering the electric conductivity new strategies for the poling process can be developed in order to improve the bending actuation. In this way, we show the importance of considering charge transportation processes in simulations of the poling of ferroelectrics, which seems not to have been recognized so far.

Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations

Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline micro- or rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples.

Transient bioheat simulation of the laser-tissue interaction in human skin using hybrid finite element formulation.

This paper presents a hybrid finite element model for describing quantitatively the thermal responses of skin tissue under laser irradiation. The model is based on the boundary integral-based finite element method and the Pennes bioheat transfer equation. In this study, temporal discretization of the bioheat system is first performed and leads to the well-known modified Helmholtz equation. A radial basis function approach and the boundary integral based finite element method are employed to obtain particular and homogeneous solutions of the laser-tissue interaction problem. In the boundary integral based finite element formulation, two independent fields are assumed: intra-element field and frame field. The intra-element field is approximated through a linear combination of fundamental solutions at a number of source points outside the element domain. The frame temperature field is expressed in terms of nodal temperature and the corresponding shape function. Numerical examples are considered to verify and assess the proposed numerical model. Sensitivity analysis is performed to explore the thermal effects of various control parameters on tissue temperature and to identify the degree of burn injury due to laser heating.

Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations

Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline microor rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples.

Nonlinear Heat Conduction in Materials with Temperature-Dependent Thermal Conductivity Using Hybrid Finite Element Model

The boundary-type hybrid finite element formulation is established to solve nonlinear heat conduction in solids with temperature-dependent material definition. The Kirchhoff transformation is first used to convert the nonlinear heat transfer system into a linear system, which is then solved by the presented hybrid finite element model, in which the fundamental solutions are used to approximate the element interior fields and conventional shape functions are used for the element boundary fields. The weak integral functional is developed to link these two fields and establish the stiffness equation. The accuracy and stability of the algorithm are tested on two examples involving various thermal conductivities to verify the present formulation.

Special fiber elements for thermal analysis of fiber‐reinforced composites

PurposeThe purpose of this paper is to present a new special element model for thermal analysis of composites.Design/methodology/approachA hybrid finite element formulation taking the fundamental solution as kernel function is presented in this work for analyzing the thermal behavior and predicting the effective thermal conductivity of fiber‐reinforced composites. A representative volume cell containing single or multiple fibers (or inclusions) is considered to investigate the overall temperature distribution affected by the inclusions and the interactions among them, and to evaluate the effective thermal conductivity of the composites using the presented algorithm with special‐purpose inclusion elements. Numerical examples are presented to demonstrate the accuracy and applicability of the proposed method in analyzing fiber‐reinforced composites.FindingsThe independent intra‐element field and frame field, as well as the newly‐developed hybrid functional, make the algorithm versatile in terms of element construction, with the result that the related variational functional involves the element boundary integral only. All numerical results are compared with the solutions from ABAQUS and good agreement is observed for all cases, clearly demonstrating the potential applications of the proposed approach to large‐scale modeling of fiber‐reinforced composites. The usage of special inclusion element can significantly reduce model meshing effort and computing cost, and simultaneously avoid mesh regeneration when the fiber volume fraction is changed.Practical implicationsDue to the fact that the established special elements exactly satisfy the interaction of matrix and fiber within the element, only element boundary integrals are involved, thus the algorithm can significantly reduce modeling effort and computing cost with less elements, and simultaneously avoid mesh regeneration when the fiber volume fraction is changed.Originality/valueBased on the special fundamental solution, a newly‐constructed inclusion element is applied to a number of test problems involving unit RVCs with multiple fibers to access the accuracy of the model. The effective thermal conductivity of the composites is evaluated for cases of single and multiple fibers using the average temperatures at certain points on a data‐collection surface. A new algorithm for evaluating effective properties with special elements is presented.

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.

Efficient numerical analysis of bone remodelling

This paper presents a formulation for the three-dimensional numerical simulation of mechanically regulated bone adaptation. Attention is focussed on a phenomenologically-based approach to bone remodelling that can be used as a computationally efficient tool to provide insight into the overall response of bone to mechanical loading. A discretisation approach is developed based on a hybrid finite element formulation where displacement, stress and density fields are approximated independently. The paper also discusses a solution algorithm tailored for shared memory multi-core computers. The performance of the model is demonstrated by two numerical examples.

Hybrid Finite Element Formulation for Electrostrictive Materials: Dynamic Analysis of Beam

A nonlinear electromechanical coupled dynamic finite element formulation for electrostrictive materials is proposed. This formulation includes the quadratic dependence of strain with polarization, valid at a constant temperature and excludes hysteresis. The present formulation uses linear finite element analysis along with the numerical solution of the nonlinear constitutive equation using Newton-Raphson technique only within each electrostrictive elements. Therefore this formulation is capable of handling nonlinearity only on the electrostrictive domain instead of making the whole finite element analysis as nonlinear. The Newton-Raphson technique is specially modified in order to guarantee the convergence of the solution. A simple technique for obtaining the initial guess of the solution for Newton-Raphson technique is also proposed which gives faster convergence of the solution. The assumption made in most of the finite element formulations, that polarization is approximately equal to electric displacement has been relaxed. The proposed finite element formulation is computationally very efficient than any other available nonlinear dynamic finite element formulation for electrostrictive materials and it proves its very high computational efficiency especially in case of electrostrictive patches embedded in large structures.

Hybrid Finite Element Formulation for Temperature Prediction in Carbon Nanofiber Based Composites

The temperature distribution in nanoscale cylindrical representative volume element (RVE) with centered carbon nanofiber (CNF) was modeled by the developed hybrid finite element approach, which used the axisymmetric Green’s function to approximate the interior field within the element, while the independent element boundary field with conventional shape functions was assumed to keep the continuity of field variables between adjacent elements. As a result, coupling with the constructed hybrid functional to link the independent interior and boundary fields, the solving linear system of equations with boundary integrals only were obtained. Numerical examples were given to show the accuracy of the proposed approach, and results were found in good agreement with those of ABAQUS. Also, it was obvious that the surface temperature of nanofiller almost kept constant during heat transfer, due to the large difference of thermal properties between nanofiller and matrix.

Fundamental-solution-based finite element model for plane orthotropic elastic bodies

A new hybrid finite element formulation is presented for solving two-dimensional orthotropic elasticity problems. A linear combination of fundamental solutions is used to approximate the intra-element displacement fields and conventional shape functions are employed to construct elementary boundary fields, which are independent of the intra-element fields. To establish a linkage between the two independent fields and produce the final displacement-force equations, a hybrid variational functional containing integrals along the elemental boundary only is developed. Results are presented for four numerical examples including a cantilever plate, a square plate under uniform tension, a plate with a circular hole, and a plate with a central crack, respectively, and are assessed by comparing them with solutions from ABAQUS and other available results.

Finite Element Method Applied to Supersonic Flutter of Circular Cylindrical Shells

DOI: 10.2514/1.39580 The method of analysis is a combination of Sander’s thin shell theory and the classic finite element method, in which the nodal displacements are found from the exact solution of shell governing equations rather than approximatedbypolynomialfunctions.Pistontheorywithandwithoutacorrectionfactorforcurvatureisappliedto derive aerodynamic damping and stiffness matrices. The influence of stress stiffness due to internal pressure and axial loading is also taken into account. Aeroelastic equations in hybrid finite element formulation are derived and solved numerically. Different boundary conditions of the shell, geometries, and flow parameters are investigated. In all study cases, the shell loses its stability due to coupled-mode flutter and a traveling wave is observed during this dynamicinstability.Theresultsarecomparedwithexistingexperimentaldataandotheranalyticaland finiteelement solutions. The present study shows efficient and reliable results that can be applied to the aeroelastic design and analysis of shells of revolution in aerospace vehicles. Nomenclature � A� = coefficient matrix of shape functions; see Appendix B.1 a1 = freestream speed of sound � B� = coefficient matrix of strain vector; see Eq. (11) � Cf� = global aerodynamic damping matrix � cf�

Hybrid finite element method applied to supersonic flutter of an empty or partially liquid-filled truncated conical shell

In this study, aeroelastic analysis of a truncated conical shell subjected to the external supersonic airflow is carried out. The structural model is based on a combination of linear Sanders thin shell theory and the classic finite element method. Linearized first-order potential (piston) theory with the curvature correction term is coupled with the structural model to account for pressure loading. The influence of stress stiffening due to internal or external pressure and axial compression is also taken into account. The fluid-filled effect is considered as a velocity potential variable at each node of the shell elements at the fluid–structure interface in terms of nodal elastic displacements. Aeroelastic equations using the hybrid finite element formulation are derived and solved numerically. The results are validated using numerical and theoretical data available in the literature. The analysis is accomplished for conical shells of different boundary conditions and cone angles. In all cases the conical shell loses its stability through coupled-mode flutter. This proposed hybrid finite element method can be used efficiently for design and analysis of conical shells employed in high speed aircraft structures.

Hybrid FEM with fundamental solutions as trial functions for heat conduction simulation

A new type of hybrid finite element formulation with fundamental solutions as internal interpolation functions, named as HFS-FEM, is presented in this paper and used for solving two dimensional heat conduction problems in single and multi-layer materials. In the proposed approach, a new variational functional is firstly constructed for the proposed HFS-FE model and the related existence of extremum is presented. Then, the assumed internal potential field constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration is used as the internal interpolation function, which analytically satisfies the governing equation within each element. As a result, the domain integrals in the variational functional formulation can be converted into the boundary integrals which can significantly simplify the calculation of the element stiffness matrix. The independent frame field is also introduced to guarantee the inter-element continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations. The proposed method inherits the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional finite element method (FEM) and boundary element method (BEM), and avoids the difficulty in selecting appropriate terms of T-complete functions used in HT-FEM, as the fundamental solutions contain usually one term only, rather than a series containing infinitely many terms. Further, the fundamental solutions of a problem are, in general, easier to derive than the T-complete functions of that problem. Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show good numerical accuracy and remarkable insensitivity to mesh distortion.

Nonlocal damage theory in hybrid-displacement formulations

This paper discusses three hybrid-displacement finite element formulations for the simulation of strain localization based on nonlocal damage theory. An isotropic integral nonlocal damage model is chosen. The hybrid finite element formulations adopted in this work are developed from first principles of Mechanics. The first one defines the domain approximations using the Trefftz functions derived for the linear elastic regime. When damage appears the hybrid-Trefftz displacement formulation degenerates into an hybrid-displacement formulation. The second formulation uses an enriched Trefftz basis with the consideration of local Heaviside functions. The third formulation uses orthonormal Legendre polynomials for the domain approximations. A set of benchmark tests is presented and discussed in order to compare the performance and accuracy of the different models. It is shown that the proposed hybrid-Trefftz formulation allows the reproduction of the general behavior of the structure but does not lead to a correct simulation of the strain tensor evolution. The hybrid-displacement formulation that uses orthonormal Legendre polynomials gives coherent results, so it appears to be a promising field of investigation.

An Efficient Lumped Mixed Hybrid Finite Element Formulation for Variably Saturated Groundwater Flow

Accurate numerical simulation of infiltration in the vadose zone remains a challenge, especially when very sharp fronts are modeled. In this work, we used the mixed hybrid finite element (MHFE) method, which allows a simultaneous approximation of both pressure head and velocity and can handle general irregular grids with highly heterogeneous permeability. For many problems dealing with unsaturated water flow, however, the MHFE solutions exhibit significant unphysical oscillations. To avoid this phenomenon, we developed an efficient mass‐lumping scheme with the MHFE method for solving the mixed form of the Richards equation. In this work, the standard and lumped MHFE formulations were compared and the ability of the lumped formulation to reduce unphysical oscillations was determined for one‐ and two‐dimensional infiltration problems. Theoretical analysis based on the M‐matrix property, which guarantees the discrete maximum principle, and practical test cases performed in this study underline the advantage of using an acute triangulation to completely remove the unphysical oscillations. Indeed, contrary to the standard approach, the lumped formulation satisfies the M‐matrix property without any constraint on the time step size to be used.

A generalized Richards equation for surface/subsurface flow modelling

Summary An approach for modelling surface/subsurface flow and transport in a fully integrated way is presented. The diffusive wave approximation used to model runoff is formulated as a nonlinear Darcy law and coupled with the Richards equation for variably saturated flow. The water dynamics in the three physical domains, land surface, vadose zone and saturated zone, is described through a single Richards type equation with domain-dependent parameters. This approach naturally provides pressure and fluxes continuity along land surface. An advective–diffusive transport equation is incorporated in the model in order to follow and identify pre-event and event water. This multi-domain Richards equation is solved with a mixed hybrid finite element formulation. The time discretization is implicit and the nonlinear equations are solved within a sequential iterative Picard scheme. This model can describe both infiltration excess or saturation excess runoff. In order to assess this modelling approach, several classical validation, verification and application test cases are presented. For overland flow solely, the model is compared to an analytical solution and to commonly used hydrological models. The integrated model is then validated with a sandbox laboratory experiment. Finally, the dynamic response of a saturated contributing area at the toe of a 3D virtual hillslope is simulated to illustrate the applicability of the model to 3D situations.

Numerical simulations and experimental measurements of the stress intensity factor in perforated plates

A numerical procedure, which combines two hybrid finite element formulations, was developed to analyse the stress intensity factors in cracked perforated plates with a periodic distribution of holes and square representative volume elements. The accuracy of the method in predicting the stress intensity factor was verified by a comparison with experimental measurements, carried out by a photoelasticity method, and by commercial finite element software. Several simulations were executed by varying both the crack length and the hole diameters, and the effects of the holes on the stress intensity factor are illustrated. The method shows high accuracy and efficiency, as small differences were observed when compared with the traditional finite element method, notwithstanding a strong reduction in degrees of freedom and mesh complexity.