A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method

This paper introduces a new algorithm to define a stable Lagrange multiplier space to impose stiff interface conditions within the context of the extended finite element method. In contrast to earlier approaches, we do not work with an interior penalty formulation as, e.g. for Nitsche techniques, but impose the constraints weakly in terms of Lagrange multipliers. Roughly speaking a stable and optimal discrete Lagrange multiplier space has to satisfy two criteria: a best approximation property and a uniform inf–sup condition. Owing to the fact that the interface does not match the edges of the mesh, the choice of a good discrete Lagrange multiplier space is not trivial. Here we propose a new algorithm for the local construction of the Lagrange multiplier space and show that a uniform inf–sup condition is satisfied. A counterexample is also presented, i.e. the inf–sup constant depends on the mesh‐size and degenerates as it tends to zero. Numerical results in two‐dimensional confirm the theoretical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

Mixed Arlequin method for multiscale poromechanics problems

An Arlequin poromechanics model is introduced to simulate the hydro‐mechanical coupling effects of fluid‐infiltrated porous media across different spatial scales within a concurrent computational framework. A two‐field poromechanics problem is first recast as the twofold saddle point of an incremental energy functional. We then introduce Lagrange multipliers and compatibility energy functionals to enforce the weak compatibility of hydro‐mechanical responses in the overlapped domain. To examine the numerical stability of this hydro‐mechanical Arlequin model, we derive a necessary condition for stability, the twofold inf–sup condition for multi‐field problems, and establish a modified inf–sup test formulated in the product space of the solution field. We verify the implementation of the Arlequin poromechanics model through benchmark problems covering the entire range of drainage conditions. Through these numerical examples, we demonstrate the performance, robustness, and numerical stability of the Arlequin poromechanics model. Copyright © 2016 John Wiley & Sons, Ltd.     

SGBEM with Lagrange multipliers applied to elastic domain decomposition problems with curved interfaces using non‐matching meshes

An original approach to the solution of linear elastic domain decomposition problems by the symmetric Galerkin boundary element method is developed. The approach is based on searching for the saddle‐point of a new potential energy functional with Lagrange multipliers. The interfaces can be either straight or curved, open or closed. The two coupling conditions, equilibrium and compatibility, along an interface are fulfilled in a weak sense by means of Lagrange multipliers (interface displacements and tractions), which enables non‐matching meshes to be used at both sides of interfaces between subdomains. The accuracy and robustness of the method is tested by several numerical examples, where the numerical results are compared with the analytical solution of the solved problems, and the convergence rates of two error norms are evaluated for h‐refinements of matching and non‐matching boundary element meshes. Copyright © 2010 John Wiley & Sons, Ltd.     

On the evaluation of the method of finite spheres for the solution of Reissner–Mindlin plate problems using the numerical inf–sup test

In this paper we use the numerical inf–sup test to evaluate both displacement‐based and mixed discretization schemes for the solution of Reissner–Mindlin plate problems using the meshfree method of finite spheres. While an analytical proof of whether a discretization scheme passes the inf–sup condition is most desirable, such a proof is usually out of reach due to the complexity of the meshfree approximation spaces involved. The numerical inf–sup test (Int. J. Numer. Meth. Engng 1997; 40 :3639–3663), developed to test finite element discretization spaces, has therefore been adopted in this paper. Tests have been performed for both regular and irregular nodal configurations. While, like linear finite elements, pure displacement‐based approximation spaces with linear consistency do not pass the inf–sup test and exhibit shear locking, quadratic discretizations, unlike quadratic finite elements, pass the test. Pure displacement‐based and mixed approximation spaces that pass the numerical inf–sup test exhibit optimal or near optimal convergence behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

Meshless methods in dual analysis: Theoretical and implementation issues

This paper presents a meshless implementation of dual analysis for 2D linear elasticity problems. The derivation of the governing systems of equations for the discretized compatible and equilibrated models is detailed and crucial implementation issues of the proposed algorithm are discussed: (i) arising of deficiencies associated with the independent approximation field used for the imposition of the essential boundary conditions (EBC) for the two parts of the boundary sharing a corner and (ii) determination of the Lagrange multipliers functional space used to impose EBC. An attempt to implement the latter resulted in an approximation which is nothing more than the trace on the essential boundary of the domain nodal functions. The difficulties posed by such approximation are explained using the inf–sup condition.Several examples of global (energy) and local (displacements) quantities of interest and their bounds determination are used to demonstrate the validity of the presented meshless approach to dual analysis. Numerical assessment of the convergence rates obtained for both models is made, for different polynomial basis degrees.     

Stable linear traction‐based equilibrium elements for elastostatics: Direct access to linear statically admissible stresses and quadratic kinematically admissible displacements for dual analysis

This paper establishes the basic framework for the traction‐based equilibrium finite element method (traction‐based EFEM). Stable linear traction‐based equilibrium elements are formulated using the macro‐element technique. An arbitrary internal macro‐point renders a linear triangular element stable, while a stable linear quadrilateral element requires the macro‐point to locate at the intersection of diagonals. Then, a Lagrangian formulation is utilized to minimize the complementary energy under equilibrium constraints, and consequently, tractions as well as additional Lagrange multipliers are obtained. Linear statically admissible (SA) stresses are thereafter acquired from tractions. As for Lagrange multipliers, they turn out to coincide well with rigid‐body displacements in each element after simple modifications. With rigid‐body displacements and linear tractions known, quadratic displacements and the kinematically admissible (KA) counterpart thereof by recovery are obtainable. The knowledge of both SA stresses and KA displacements renders dual analysis directly applicable. That is to say, the traction‐based EFEM is featured with direct access to strict upper and lower bounds of strain energy and other quantities of interest. Copyright © 2014 John Wiley & Sons, Ltd.     

Fast BEM–FEM mortar coupling for acoustic–structure interaction

A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure–acoustic field interaction. Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain. The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non‐matching grids. The resulting saddle‐point problem is solved by an approximate Uzawa‐type scheme in which the matrix–vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method. The algorithm is demonstrated on the example of a cavity‐backed elastic panel. Copyright © 2005 John Wiley & Sons, Ltd.     

巴西圆盘中心裂纹摩擦接触问题的逐点Lagrange乘子法

采用逐点Lagrange乘子法求解巴西圆盘中心裂纹在压剪荷载作用下裂纹面可能发生的摩擦接触问题。为了避免传统的Lagrange乘子法中总刚度阵求逆的困难,将Lagrange乘子逐点转到局部坐标系下,采用Gauss-Seidel迭代法求解法向和切向乘子,同时注意在求解的过程中对切向乘子约束修正,待所有点乘子求解完成后再变换到整体坐标系下迭代求解位移。与传统接触算法相比,该算法无需对总刚度阵求逆,降低了求解规模,提高了计算效率。通过该方法计算了巴西圆盘中心裂纹两种典型情况下的应力强度因子,计算结果与文献比较,吻合良好。考虑不同荷载角和裂纹长度对位移,应力强度因子和接触区的影响,并对不同摩擦系数下应力强度因子的影响进行了分析。结果表明：忽略裂纹接触摩擦作用,应力强度因子可能被高估。     

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.     

Reduction of substructural interface degrees of freedom in flexibility‐based component mode synthesis

A flexibility‐based component mode synthesis (CMS) is proposed for reduced‐order modelling of dynamic behaviour of large structures. The approach employs partitioning via the localized Lagrange multiplier method. The use of the localized Lagrange multipliers leads to, unlike the classical Lagrange multipliers, a linearly independent set of interface forces without any redundancies at multiply connected interface nodes. The flexibility‐based CMS method has shown significant advantages over the classical Craig–Bampton method. A key feature of the method is its substructural mode selection criterion that is independent of loading conditions. Unlike the majority of available CMS approaches, where one retains the full dimension of partition boundary degrees of freedom (DOFs), the flexibility‐based method allows to reduce significantly the interface DOFs. The reduction of interface DOFs represents the major contribution of the present communication. The efficiency of the proposed approach is demonstrated on an analysis of a simple plate partitioned and of a more complex 3D structure, both partitioned into several substructures. Copyright © 2006 John Wiley & Sons, Ltd.     

Locally constrained projections on grids

A technique is formulated for projecting vector fields from one unstructured computational grid to another grid so that a constraint condition such as a conservation property holds at the cell or element level on the ‘receiving’ grid. The approach is based on ideas from constrained optimization and certain mixed or multiplier‐type finite element methods in which Lagrange multipliers are introduced on the elements to enforce the constraint. A theoretical analysis and estimates for the associated saddle‐point problem are developed and a new algorithm is proposed for efficient solution of the resulting discretized problem. In the algorithm a reduced Schur's complement problem is constructed for the multipliers and the projected velocity computation reduces to a post‐processing calculation. In some instances the reduced system matrix can be factored so that repeated projections involve little more than forward and backward substitution sweeps. Numerical tests with an element of practical interest demonstrate optimal rate of convergence for the projected velocities and verify the local conservation property to expected machine precision. A practical demonstration for environmental simulation of Florida Bay concludes the study. Copyright © 2001 John Wiley & Sons, Ltd.     

A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers

A nonoverlapping domain decomposition (DD) method is proposed for the iterative solution of systems of equations arising from the discretization of Helmholtz problems by the discontinuous enrichment method. This discretization method is a discontinuous Galerkin finite element method with plane wave basis functions for approximating locally the solution and dual Lagrange multipliers for weakly enforcing its continuity over the element interfaces. The primal subdomain degrees of freedom are eliminated by local static condensations to obtain an algebraic system of equations formulated in terms of the interface Lagrange multipliers only. As in the FETI‐H and FETI‐DPH DD methods for continuous Galerkin discretizations, this system of Lagrange multipliers is iteratively solved by a Krylov method equipped with both a local preconditioner based on subdomain data, and a global one using a coarse space. Numerical experiments performed for two‐ and three‐dimensional acoustic scattering problems suggest that the proposed DD‐based iterative solver is scalable with respect to both the size of the global problem and the number of subdomains. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust imposition of Dirichlet boundary conditions on embedded surfaces

We develop both stable and stabilized methods for imposing Dirichlet constraints on embedded, three‐dimensional surfaces in finite elements. The stable method makes use of the vital vertex algorithm to develop a stable space for the Lagrange multipliers together with a novel discontinuous set of basis functions for the multiplier field. The stabilized method, on the other hand, follows a Nitsche type variational approach for three‐dimensional surfaces. Algorithmic and implementational details of both methods are provided. Several three‐dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches. The results indicate that both methods yield optimal rates of convergence in various quantities of interest, with the primary differences being in the surface flux. The utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Automatically inf − sup compliant diamond‐mixed finite elements for Kirchhoff plates

We develop a mixed finite‐element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment‐equilibrium problem for the rotations is in direct analogy to the problem of incompressible two‐dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf ? sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment‐equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf ? sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf ? sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A domain coupling method for finite element digital image correlation with mechanical regularization: Application to multiscale measurements and parallel computing

A general method is proposed to couple two subregions analyzed with finite element digital image correlation even when using a mechanical regularization (regularized digital image correlation). A Lagrange multiplier is introduced to stitch both displacements fields in order to recover continuity over the full region of interest. Another interface unknown is introduced to ensure, additionally, the equilibrium of the mechanical models used for regularization. As a first application, the method is used to perform a single measurement from images at two different resolutions. Secondly, the method is also extended to parallel computing in regularized digital image correlation. The problem is formulated at the interface and solved with a Krylov‐type algorithm. A dedicated preconditioner is proposed to significantly accelerate convergence. The resulting method is a good candidate for the analysis of large data sets. Copyright © 2016 John Wiley & Sons, Ltd.     

On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method

We consider a problem stemming from recent models of phase transitions in stimulus‐responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly ‘fit’ the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain‐integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

In this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress‐driven and strain‐driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite‐element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill–Mandel principle of macro‐homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated ‘forces’ that are conjugate to the macroscopic strain ‘displacements’. In contrast to most homogenization schemes, the second Piola–Kirchhoff stress and Green–Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola–Kirchhoff stress and the deformation gradient is discussed in the Appendix. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of Particle Methods to Static Fracture of Reinforced Concrete Structures

Particle methods for modeling reinforced concrete are described. The reinforcements are modeled by finite elements and are coupled to the particle method by Lagrange multipliers. The method is applicable to nonlinear problems, problems with moderate to severe cracking and deformable interfaces. Applications to the static response of reinforced concrete structures where the concrete is discretized with particles and the reinforcement with elements are described. The method is also tested for several static problems where no relative displacements between the concrete and the reinforcement are allowed.