An energy–momentum consistent method for transient simulations with mixed finite elements developed in the framework of geometrically exact shells

In this paper, a mixed variational formulation for the development of energy–momentum consistent (EMC) time‐stepping schemes is proposed. The approach accommodates mixed finite elements based on a Hu–Washizu‐type variational formulation in terms of displacements, Green–Lagrangian strains, and conjugated stresses. The proposed discretization in time of the mixed variational formulation under consideration yields an EMC scheme in a natural way. The newly developed methodology is applied to a high‐performance mixed shell finite element. The previously observed robustness of the mixed finite element formulation in equilibrium iterations extends to the transient regime because of the EMC discretization in time. Copyright © 2016 John Wiley & Sons, Ltd.     

Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid

This paper presents the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. The originality of this work lies (i) in the development of a variational formulation for the fully coupled fluid/piezoelectric structure system, and (ii) in the finite element implementation of an inexpensive and accurate axisymmetric adaptive laminated conical shell element. Various modal results are presented in order to validate and illustrate the efficiency of the proposed fluid–structure finite element formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

BOUNDARY ELEMENT–FINITE ELEMENT COUPLED EIGENANALYSIS OF FLUID–STRUCTURE SYSTEMS

The eigenanalysis of acoustical cavities with flexible structure boundaries, such as a fluid-filled container or an automobile cabin enclosure, is considered. An algebraic eigenvalue problem formulation for the fluid–structure problem is presented by combining the acoustic fluid boundary element eigenvalue analysis method and the structural finite elements. For many practical eigenproblems, use of finite elements to discretize the fluid domain leads to large stiffness and mass matrices. Since the acoustic boundary element discretization requires putting nodes only on the wetted surface of the structure, the size of the eigenproblem is reduced considerably, thus reducing the eigenvalue extraction effort. Futhermore, unlike in ordinary cases, the finite element discretization of pressure–displacement based fluid–structure problem gives rise to unsymmetric matrices. Therefore, the fact that the boundary element formulation produces unsymmetric matrices does not introduce additional difficulties here compared to the finite element case in the choice of an eigenvalue extraction procedure. Examples are included to demonstrate the fluid–structure eigenanalysis using boundary elements for the fluid domain and finite elements for the structure.     

On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems

This paper describes variational formulation and finite element discretization of surface tension. The finite element formulation is cast in the Lagrangian framework, which describes explicitly the interface evolution. In this context surface tension formulation emerges naturally through the weak form of the Laplace–Young equation.The constitutive equations describing the behaviour of Newtonian fluids are approximated over a finite time step, leaving the governing equations for the free surface flow function of geometry change rather than velocities. These nonlinear equations are then solved by using Newton-Raphson procedure.Numerical examples have been executed and verified against the solution of the ordinary differential equation resulting from a parameterization of the Laplace-Young equation for equilibrium shapes of drops and liquid bridges under the influence of gravity and for various contact angle boundary conditions.     

A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator

The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.     

A new acoustic interface element for fluid-structure interaction problems

A new comprehensive acoustic 2-D interface element capable of coupling the boundary element (BE) and finite element (FE) discretizations has been formulated for fluid–structure interaction problems. The Helmholtz equation governing the acoustic pressure in a fluid is discretized using the BE method and coupled to the FE discretization of a vibrating structure that is in contact with the fluid. Since the BE method naturally maps the infinite fluid domain into finite node points on the fluid–structure interface, the formulation is especially useful for problems where the fluid domain extends to infinity. Details of the BE matrix computation process adapted to FE code architecture are included for easy incorporation of the interface element in FE codes. The interface element has been used to solve a few simple fluid–structure problems to demonstrate the validity of the formulation. Also, the vibration response of a submerged cylindrical shell has been computed and compared with the results from an entirely finite element formulation.     

The variational formulation and solution of problems of finite-strain elastoplasticity based on the use of a dissipation function

The solution of initial-boundary value problems involving finite elastoplastic deformations is discussed. The formulation considered differs from conventional formulations in that the evolution law is expressed in terms of the dissipation function. A generalized midpoint rule is used to obtain an incremental problem, a variational form of which is derived. The finite element method is used for spatial discretization, and an algorithm to solve the resulting discrete problem is developed. This algorithm has the predictor–corrector structure common to most solution procedures for problems in plasticity. Methods for imposing the plastic incompressibility constraint are investigated. Solutions to two axisymmetric examples obtained using the proposed algorithm are presented and compared with those obtained by other authors.     

Smart element method I. The Zienkiewicz–Zhu feedback

A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.     

A new finite element formulation for internal acoustic problems with dissipative walls

This work concerns the variational formulation and the numerical computation of internal acoustic problems with absorbing walls. The originality of the proposed approach, compared to other existing methods, is the introduction of the normal fluid displacement field on the damped walls. This additional variable allows to transpose formulations in frequency domain to time domain when the fluid is described by a scalar field (pressure or fluid displacement potential). With this new scalar unknown, various absorbing wall models can be introduced in the variational formulation. Moreover, the associated finite element matrix system in symmetric form can be solved in frequency and time domain. Copyright © 2006 John Wiley & Sons, Ltd.     

Genuinely resultant shell finite elements accounting for geometric and material non-linearity

A general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

超弹性柔性结构与流体耦合运动的浸入边界法研究

浸入边界法是模拟大变形柔性弹性结构和粘性流体相互作用的重要数值方法之一。该文有效结合传统的反馈力方法和混合有限元浸入边界方法,对圆柱和方柱绕流后柔性悬臂梁流固耦合振动问题进行数值模拟。其中,固体采用超弹性材料,利用有限单元法求解,流体为不可压缩牛顿流体,使用笛卡尔自适应加密网格,利用有限差分法进行求解。通过数值计算,得到柔性超弹性结构的耦合振动特性和流场动态分布特性,并将计算结果同其他文献计算结果进行比较,验证了该耦合计算方法的可靠性。     

Lippmann-Schwinger solvers for the explicit jump discretization for thermal computational homogenization problems

We present a variational formulation and a Lippmann-Schwinger equation for the explicit jump discretization of thermal computational homogenization problems, together with fast and memory-efficient matrix-free solvers based on the fast Fourier transform (FFT). Wiegmann and Zemitis introduced the explicit jump discretization for volumetric image-based computational homogenization of thermal conduction. In contrast to Fourier and finite difference-based discretization methods classically used in FFT-based homogenization, the explicit jump discretization is devoid of ringing and checkerboarding artifacts. Originally, the explicit jump discretization was formulated as the discrete equivalent of a boundary integral equation for the jump in the temperature gradient. The resulting equations are not symmetric positive definite, and thus solved by the BiCGSTAB method. Still, the numerical scheme exhibits stable convergence behavior, also in the presence of pores. In this work, we exploit a reformulation of the explicit jump system in terms of harmonically averaged conductivities. The resulting system is intrinsically symmetric positive definite and admits a Lippmann-Schwinger formulation. A seamless integration into existing FFT-based software packages is ensured. We demonstrate our improvements by numerical experiments.     

A multiscale stabilized ALE formulation for incompressible flows with moving boundaries

This paper presents a variational multiscale stabilized finite element method for the incompressible Navier–Stokes equations. The formulation is written in an Arbitrary Lagrangian–Eulerian (ALE) frame to model problems with moving boundaries. The structure of the stabilization parameter is derived via the solution of the fine-scale problem that is furnished by the variational multiscale framework. The projection of the fine-scale solution onto the coarse-scale space leads to the new stabilized method. The formulation is integrated with a mesh moving scheme that adapts the computational grid to the evolving fluid boundaries and fluid-solid interfaces. Several test problems are presented to show the accuracy and stability of the new formulation.     

Finite elements in space and time for generalized viscoelastic maxwell model

 The generalization of a new numerical approach with simultaneous space–time finite element discretization for viscoelastic problems developed in the papers by Buch et al. (1999) and Idesman et al. (2000) is presented for the case of the generalized viscoelastic Maxwell model. New non-symmetric variational and discretized formulations are derived using the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour described by the generalized Maxwell model is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations with total displacements and internal displacements (internal variables) as unknowns, namely to the equilibrium equation and the evolution equations for the internal displacements which are fulfilled in the weak form. Using continuous test functions in space and time, a continuous space–time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into appropriate time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across time interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space–time refinements and efficient parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space–time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time and for solutions with high gradients. Received 7 February 2000     

Finite volume discretization for poroelastic media with fractures modeled by contact mechanics

A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.     

Mixed-variational formulation for phononic band-structure calculation of arbitrary unit cells

This paper presents phononic band-structure calculation results obtained using a mixed variational formulation for 1-, and 2-dimensional unit cells. The formulation itself is presented in a form which is equally applicable to 3-dimensiomal cases. It has been established that the mixed-variational formulation presented in this paper shows faster convergence with considerably greater accuracy than variational principles based purely on the displacement field, especially for problems involving unit cells with discontinuous constituent properties. However, the application of this formulation has been limited to fairly simple unit cells. In this paper we have extended the scope of the formulation by employing numerical integration techniques making it applicable for the evaluation of the phononic band-structure of unit cells displaying arbitrary complexity in their Bravais structure and in the shape, size, number, and anisotropicity of their micro-constituents. The approach is demonstrated through specific examples.     

An energy-stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor-Hood element based on nonuniform rational B-splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf-sup condition is utilized to provide a bound for the pressure field. The generalized-α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf-sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.     

A coupled FEM-MFS method for the vibro-acoustic simulation of laminated poro-elastic shells

A new simulation method for the vibro-acoustic simulation of poro-elastic shells is presented. The proposed methods can be used to investigate arbitrary curved layered panels, as well as their interaction with the surrounding air. We employ a high-order finite element method (FEM) for the discretization of the shell structure. We assume that the shell geometry is given parametrically or implicitly. For both cases the exact geometry is used in the simulation. In order to discretize the fluid surrounding the structure, a variational variant of the method of fundamental solutions (MFS) is developed. Thus, the meshing of the fluid domain can be avoided and in the case of unbounded domains the Sommerfeld radiation condition is fulfilled. In order to simulate coupled fluid-structure interaction problems, the FEM and the MFS are combined to a coupled method. The implementation of the uncoupled FEM for the shell and the uncoupled MFS is verified against numerical examples based on the method of manufactured solutions. For the verification of the coupled method an example with a known exact solution is considered. In order to show the potential of the method sound transmission from cavities to exterior half-spaces is simulated.     

Sequential quadratic programming for non-linear elastic contact problems

The physical problem considered in this paper is that of a non-linear elastic body being indented by a rigid punch. The treatment is based on finite element discretization and sequential quadratic programming (SQP). The finite element formulation is obtained through a variational formulation, which generalizes to frictionless contact a three-field principle which involves deformation, volume strain and hydrostatic pressure as independent fields. We compare an incremental load method and a method where the indentation for the final load is sought directly. Crucial for the second method is the use of a line search with respect to a merit function which measures the infeasibility in the optimality criteria for the problem; this line search also includes a check of the orientation-preserving condition of a positive determinant of the deformation gradient. Each iteration within an SQP method requires the solution of a quadratic programming (QP) subproblem, and four different methods for the solution of these subproblems are compared. The performance of the overall procedure is also compared to that of a commercially available system. Test examples ranging from 23 to 770 displacement degrees of freedom are treated. The computational results show that the proposed solution concept is feasible and efficient. Furthermore, it can be applied to general non-linear elastic contact problems, since it does not include any ad hoc rules.