Localized Lagrange Multipliers Research Articles

Analysis of composite material interface crack face contact and friction effects using a new node-pairs contact algorithm

A new node-pairs contact algorithm is proposed to deal with a composite material or bi-material interface crack face contact and friction problem (e.g., resistant coating and thermal barrier coatings) subjected to complicated load conditions. To decrease the calculation scale and calculation errors, the local Lagrange multipliers are solved only on a pair of contact nodes using the Jacobi iteration method, and the constraint modification of the tangential multipliers are required. After the calculation of the present node-pairs Lagrange multiplier, it is turned to next contact node-pairs until all node-pairs have finished. Compared with an ordinary contact algorithm, the new local node-pairs contact algorithm is allowed a more precise element on the contact face without the stiffness matrix singularity. The stress intensity factors (SIFs) and the contact region of an infinite plate central crack are calculated and show good agreement with those in the literature. The contact zone near the crack tip as well as its influence on singularity of stress fields are studied. Furthermore, the frictional contacts are also considered and found to have a significant influence on the SIFs. The normalized mode-II stress intensity factors K̂II for the friction coefficient decrease by 16% when f changes from 1 to 0.

국부 및 혼합 Lagrange 승수법을 이용한 영역분할 기반 유한요소 구조해석 기법 개발

In this paper, a finite element domain decomposition method using local and mixed Lagrange multipliers for a large scal structural analysis is presented. The proposed algorithms use local and mixed Lagrange multipliers to improve computational efficiency. In the original FETI method, classical Lagrange multiplier technique was used. In the dual-primal FETI method, the interface nodes are used at the corner nodes of each sub-domain. On the other hand, the proposed FETI-local analysis adopts localized Lagrange multipliers and the proposed FETI-mixed analysis uses both global and local Lagrange multipliers. The numerical analysis results by the proposed algorithms are compared with those obtained by dual-primal FETI method.

The nsBETI method: an extension of the FETI method to non‐symmetrical BEM‐FEM coupled problems

SUMMARYThe finite element tearing and interconnecting (FETI) method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second‐order elasticity problems. In the boundary element tearing and interconnecting (BETI) method, a direct extension of the FETI algorithm to the BEM, the symmetric Galerkin BEM formulation, is used to obtain symmetric system matrices, making possible to apply the same FETI conjugate gradient solver. In this work, we propose a new BETI variant labeled nsBETI that allows to couple substructures modeled with the FEM and/or non‐symmetrical BEM formulations. The method connects non‐matching BEM and FEM subdomains using localized Lagrange multipliers and solves the associated non‐symmetrical flexibility equations with a Bi‐CGstab iterative algorithm. Scalability issues of nsBETI in BEM–BEM and combined BEM–FEM coupled problems are also investigated. Copyright © 2012 John Wiley & Sons, Ltd.

Semiparametric Inference in Correlated Long Memory Signal Plus Noise Models

This article proposes an extension of the log periodogram regression in perturbed long memory series that accounts for the added noise, while also allowing for correlation between signal and noise, a common situation in many economic and financial series. Consistency (for d < 1) and asymptotic normality (for d < 3/4) are shown with the same bandwidth restriction as required for the original log periodogram regression in a fully observable series, with the corresponding gain in asymptotic efficiency and faster convergence over competitors. Local Wald, Lagrange Multiplier, and Hausman type tests of the hypothesis of no correlation between the latent signal and noise are also proposed.

Partitioned vibration analysis of internal fluid‐structure interaction problems

SUMMARYA partitioned, continuum‐based, internal fluid–structure interaction (FSI) formulation is developed for modeling combined sloshing, acoustic waves, and the presence of an initial pressurized state. The present formulation and its computer implementation use the method of localized Lagrange multipliers to treat both matching and non‐matching interfaces. It is shown that, with the context of continuum Lagrangian kinematics, the fluid sloshing and acoustic stiffness terms originate from an initial pressure term akin to that responsible for geometric stiffness effects in solid mechanics. The present formulation is applicable to both linearized vibration analysis and nonlinear FSI transient analysis provided that a convected kinematics is adopted for updating the mesh geometry in a finite element discretization. Numerical examples illustrate the capability of the present procedure for solving coupled vibration and nonlinear sloshing problems. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Equilibration techniques for solving contact problems with Coulomb friction

In this paper, we consider residual and equilibrated error indicators for contact problems with Coulomb friction. The contact problem is handled within the abstract framework of saddle point problems. More precisely, the non-penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual Lagrange multiplier space. Thus from the displacement, we can easily compute in a local post-process the Lagrange multiplier which acts as a Neumann condition on the possible contact zone. Having computed the discrete Lagrange multiplier, we can apply standard error estimators by replacing the unknown Neumann data by its approximation. As it is shown in [1], this results in an error estimator for a one-sided contact problem without friction. Here, we consider more general situations and discuss two additional contact terms which measure the non-conformity of the discrete Lagrange multiplier. Numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process.

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

We present the asynchronous multi-domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary-value problem in hyperelasticity. Variational time integration schemes, based on the principle of minimal action within the Lagrangian framework, are constructed for the equation of motion and implemented into a variational finite element framework, which is systematically derived from the three-field de Veubeke-Hu-Washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly-incompressible materials. For efficient parallel computing, we use the dual domain decomposition method with local Lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains. The α-method for time discretization and the multi-domain spatial decomposition enable us to use different types of integrators (explicit vs. implicit) and different time steps on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. The energy conservation of our nonlinear, midpoint, asynchronous integration scheme is investigated using the Energy method, and both local and the global energy error estimates are derived. We illustrate the performance of proposed variational multi-domain time integrators by means of three examples. First, the method of manufactured solutions is used to examine the consistency of the formulation. In the second example, we investigate energy conservation and stability. Finally, we apply the method to the motion of a heterogeneous plane domain, where different integrators and time discretization steps are used accordingly with disparate material data of individual parts.

Partitioned formulation of internal and gravity waves interacting with flexible structures

This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures. The governing interaction model consists of three completely partitioned entities: fluid model, structural model, and interface model that acts as an internal constraint on the fluid–structure interface boundary. Thus, the proposed partitioned multi-physics modeling can employ two completely modular fluid and structure software modules plus an interface solver, hence amenable to partitioned solution algorithms. The interface discretization can exploit the nonmatching interface algorithm previously developed via the method of localized Lagrange multipliers. Also noted is that the present fluid model can make use of widely available finite element software for standard Poisson-type problems.

Parallel computation approaches for flexible multibody dynamics simulations

Finite element based formulations for flexible multibody systems are becoming increasingly popular and as the complexity of the configurations to be treated increases, so does the computational cost. It seems natural to investigate the applicability of parallel processing to this type of problems; domain decomposition techniques have been used extensively for this purpose. In this approach, the computational domain is divided into non-overlapping sub-domains, and the continuity of the displacement field across sub-domain boundaries is enforced via the Lagrange multiplier technique. In the finite element literature, this approach is presented as a mathematical algorithm that enables parallel processing. In this paper, the divided system is viewed as a flexible multibody system, and the sub-domains are connected by kinematic constraints. Consequently, all the techniques applicable to the enforcement of constraints in multibody systems become applicable to the present problem. In particular, it is shown that a combination of the localized Lagrange multiplier technique with the augmented Lagrange formulation leads to interesting solution strategies.

Partitioning based reduced order modelling approach for transient analyses of large structures

PurposeThe purpose of this paper is to describe reduced order modelling based on dynamic flexibility approximation and applied to transient analyses.Design/methodology/approachThis work is based on a recently proposed flexibility‐based component modes synthesis (CMS) approach which was shown to be very efficient for solving large eigenvalue problems. The model reduction approach is based on partionning via the localized Lagrange multipliers method, which makes it very appropriate to handle coupled problems.FindingsIn particular, it is demonstrated in this paper how the utilised model reduction method can be applied only to one part of the structure and efficiently coupled to a full finite element model. The performance of the method is investigated on numerical examples of plate and 3D problems.Originality/valueThe proposed flexibility‐based CMS approach can be used as a very efficient tool for complex engineering structures under dynamic load where the mode superposition method applies. The efficiency of the computations is brought about by the model reduction.

Treatment of acoustic fluid–structure interaction by localized Lagrange multipliers and comparison to alternative interface-coupling methods

This paper is a sequel on the topic of localized Lagrange multipliers (LLM) for applications of fluid–structure interaction (FSI) between finite-element models of an acoustic fluid and an elastic structure. The prequel paper formulated the spatial-discretization methods, the LLM interface treatment, the time-marching partitioned analysis procedures, and the application to 1D benchmark problems. Here, we expand on formulation aspects required for successful application to more realistic 2D and 3D problems. Additional topics include duality relations at the fluid–structure interface, partitioned vibration analysis, reduced-order modeling, handling of curved interface surfaces, and comparison of LLM with other coupling methods. Emphasis is given to non-matching fluid–structure meshes. We present benchmark examples that illustrate the benefits and drawbacks of competing interface treatments. Realistic application problems involving the seismic response of two existing dams are considered. These include 2D modal analyses of the Koyna gravity dam, transient-response analyses of that dam with and without reduced-order modeling, incorporation of nonlinear cavitation effects, and the 3D transient-response analysis of the Morrow Point arch dam.

Asynchronous multi‐domain variational integrators for non‐linear problems

AbstractWe develop an asynchronous time integration and coupling method with domain decomposition for linear and non‐linear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than by stability requirements. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Treatment of acoustic fluid–structure interaction by localized Lagrange multipliers: Formulation

A new concept is presented for modeling the dynamic interaction between an acoustic fluid and an elastic structure. The coupling of this multiphysics system is done by inserting a kinematic interface frame between the fluid and the structure, and using node-collocated Lagrange multipliers to connect the frame to each subsystem. The time-domain response analysis is performed by a partitioned analysis procedure. The main advantages of this localized Lagrange multiplier (LLM) primal-dual coupling method are: complete localization of the structure and fluid subsystems, elimination of the conventional predictor in the partitioned time integration method, and the ability to accommodate non-matching meshes. The standard Newmark time integrator is used on both the fluid and structure models. It is shown that if the integrator is A-stable and second-order accurate for a monolithic treatment, it retains those properties for both Mortar and LLM partitioned solution procedures. Infinite and finite piston problems are used to explain and verify the methodology. A sequel paper under preparation presents and discusses a set of benchmark and application examples that involve the response of existing dams to seismic excitation.

How paper folds: bending with local constraints

A variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry. This framework is applied to describe the patterns of a folded sheet of paper. The unstretchability of paper implies a constraint on the surface metric; bending is penalized by an energy quadratic in mean curvature. The local Lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet. The framework is illustrated by examining the deformation of a flat sheet into a generalized cone.

A formulation based on localized Lagrange multipliers for BEM–FEM coupling in contact problems

This paper presents a unified formulation for the combination of the finite element method (FEM) and the boundary element method (BEM) in 3D frictional contact problems that is based on the use of localized Lagrange multipliers (LLMs). Resolution methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids using classical Lagrange multipliers. These methods tend to generate strongly coupled systems that require a deep knowledge of the discretization characteristics on each side of the contact zone complicating the process of mixing different numerical techniques. In this work a displacement contact frame is inserted between the FE and BE interface meshes, discretized and finally connected to the contacting substructures using LLMs collocated at the mesh-interface nodes. This methodology will provide a partitioned formulation which preserves software modularity and facilitates the connection of non-matching FE and BE meshes.

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

AbstractA 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 &amp; Sons, Ltd.

FEM and BEM coupling in elastostatics using localized Lagrange multipliers

AbstractThe equations produced by the finite and boundary element methods in structural mechanics are expressed in different variables and cannot be linked without modifications. Conventional coupling methods of these numerical techniques have traditionally been based on the use of classical Lagrange multipliers making a direct connection of the two solids through their common interfaces using matching meshes and altering the formulation of one of the methods to make it compatible with the other. In this work, a discrete surface called frame is interposed between the connected subdomains to approximate their common interface displacements, it is treated using a finite element discretization and connected to each substructure using localized Lagrange multipliers collocated at the interface nodes. This methodology facilitates the connection of non‐matching finite and boundary element meshes avoiding modifications to the numerical methods used and providing a partitioned formulation, which preserves software modularity. Copyright © 2006 John Wiley &amp; Sons, Ltd.

A formulation of conserving impact system based on localized Lagrange multipliers

This paper proposes a new finite element method of frictional impact of elastic bodies. The formulation introduces a contact frame that is placed in between contacting bodies and represents the contact surface. The nonpenetration condition and the slip–stick condition are defined between the contacting body and the contact frame with the aid of the independent localized Lagrange multipliers representing the contact forces. The position of the contact frame and the local coordinate of the contacting node along the contact frame are also treated as the independent variable, which enables the exact satisfaction of the constraint conditions without the deficiency or redundant constraint. The energy and momentum conservation algorithm is applied to the proposed impact system. For the case of frictional impact, the linear momentum is exactly conserved and the angular momentum is approximately conserved with negligible error. Copyright © 2006 John Wiley & Sons, Ltd.

Partitioned formulation of frictional contact problems using localized Lagrange multipliers

AbstractInterface treatment methods for the contact problem between non‐matching meshes have traditionally been based on a direct coupling of the contacting solids employing a master–slave strategy or classical Lagrange multipliers. These methods tend to generate strongly coupled systems that is dependent on the discretization characteristics on each side of the contact zone. In this work a displacement frame is intercalated between the interface meshes. The frame is then discretized so that the discrete frame nodes are connected to the contacting substructures using the localized Lagrange multipliers collocated at the interface nodes. The resulting methodology alleviates the need for master–slave book‐keeping and provides a partitioned formulation which preserves software modularity, facilitates non‐matching mesh treatment and passes the contact patch test. Frictional contact problems are used to demonstrate the salient features of the proposed method. Copyright © 2005 John Wiley &amp; Sons, Ltd.

On micro–macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials

In this paper we introduce a two scale computational strategy for modeling the inelastic behavior of heterogeneous materials. We apply the FEM not only on the structural scale but also on the lower, so called micro scale, where it replaces the standard procedure, in which we define the material behavior through the constitutive relations. The special care is given to handling the interface condition between micro- and macro scales. The proposed approach is based on the localized Lagrange multiplier method and can lead to the realization of either displacement based or force based formulation. Numerical tests were done on porous and composite materials. We investigate the influence of the boundary conditions, micro scale representation window size and phase contrast on the heterogeneous material response of a single FE as well as of the whole structure.