Localized Lagrange Multipliers Research Articles

Unconditionally maximum principle-preserving linear method for a mass-conserved Allen–Cahn model with local Lagrange multiplier

In this work, we present a conservative Allen–Cahn (CAC) equation and investigate its unconditionally maximum principle-preserving linear numerical scheme. The operator splitting strategy is adopted to split the CAC model into a conventional AC equation and a mass correction equation. The standard finite difference method is used to discretize the equations in space. In the first step, the temporal discretization of the AC equation is performed by using the energy factorization technique. The discrete version of the maximum principle-preserving property for the AC equation is unconditionally satisfied. In the second step, we apply mass correction by using an explicit Euler-type approach. Without the constraint of time step, we estimate that the absolute value of the updated solution is bounded by 1. The unique solvability is analytically proved. In each time step, the proposed method is easy to implement because we only need to solve a linear elliptic type equation and then correct the solution in an explicit manner. Various computational experiments in two-dimensional and three-dimensional spaces are performed to confirm the performance of the proposed method. Moreover, the experiments also indicate that the proposed model can be used to simulate two-phase incompressible fluid flows.

Delay effects on distributed constrained optimization over double‐integrator multi‐agent systems

AbstractCommunication plays a pivotal role in distributed optimization problems, where unavoidable communication delays are presented. This research studies the distributed constrained optimization problem concerning second‐order multi‐agent systems with double‐integrator under time‐varying communication delays. An adaptive distributed optimization algorithm using multi‐agent system consensus technique and Karush–Kuhn–Tucker conditions is developed to deal with this problem. The local constraint term is solved adaptively through local dual Lagrange multipliers. When the cost function is strongly convex, and the communication topology is undirected and connected, we employ the Lasalle invariance principle to analyze the delay effects on convergence analysis. Moreover, we give an upper bound on communication delay. Finally, the provided numerical simulation examples demonstrate that the developed method is robust for the limited communication delay and the derived results are conservative.

Partitioned analysis of acoustic fluid–solid-saturated porous medium interaction problems by a generalized saturated porous medium model and localized Lagrange multipliers

A partitioned formulation of acoustic fluid–solid-saturated porous medium interaction problems is presented based on a generalized saturated porous medium (GSPM) model and a new localized Lagrange multiplier (LLM) method. In this formula, all various media are modeled as the GSPM and a novel, powerful continuous condition is proposed for the interaction interfaces. This condition applies not only to interfaces of two partitions but also to intersections of simultaneous interactions of any number of media with different porosities. A (u, p) frame is introduced between partitions and the LLM fields are used to connect partitions with the frame. Furthermore, finite element spatial discretization and time integration strategies are provided for both matching meshes and nonmatching meshes. This partitioned formulation is validated numerically by computing the dynamic response of a horizontally layered seawater-seabed-bedrock model subjected to plane waves. The application scenarios for this method include, but are not limited to, the dynamic coupling analysis of marine engineering structures and hydraulic engineering structures. Future work will present and discuss a set of benchmarks and applications that involve the responses of existing dams to seismic excitation.

Nonlinear Soil Behavior Model in Localized Lagrange Multipliers Mixed Formulation (u,p) for Dynamical Analysis of Wind Turbine Coupled Systems

Coupled mechanical systems can be complex, especially if there are many systems connected together and nonlinearities are present. Soil-structure interaction refers to the interaction between a structure and the soil or foundation upon which it is built. This interaction is important because it can affect the behavior of the structure, particularly during earthquakes or other dynamic events. For the wind turbine foundation design, the soil should support the weight of a wind turbine and anchor it to the ground. This paper presents a nonlinear material behavior soil coupled to elastic structure in offshore conditions. The goal of this work is to develop a coupled structural finite element procedure using Localized Lagrange Multipliers (LLM) at idealized offshore wind turbines with nonlinear poroelastic model for soil foundation. In this work, an anisotropic sand constitutive model is used to describe the soil foundation behavior. This model is based in a critical state soil mechanics and bounding surface plasticity. Classical elasto-plasticity theory is used to obtain the soil stiffness. The numerical model is validated through a fully coupled model at classical problems results. The mixed formulation [Formula: see text] is used to model the interface frames between the domains. The momentum equilibrium and mass continuity equations are solved by algebraic equation system imposed by Lagrange multipliers methodology. In order to excite the system, aerodynamic forces are imposed in random wind velocity conditions. In another numerical case, the response of coupled system during shaking is simulated by a horizontal input motion at bottom of soil foundation. Some computational aspects are discussed and the numerical model is clarified. The classical Newton Method to solve nonlinear problems is used in a finite element approach. In addition, two foundation models are tested and time dynamic responses are evaluated for a range of physical parameters including the wind nature and soil properties.

Asynchronous time integration while achieving zero interface energy

This contribution deals with an asynchronous direct time integration of the finite-element model. The proposed method is applied to the phenomenon of wave propagation through an elastic linear continuum. The numerical model is partitioned into individual subdomains using the domain decomposition method by means of localized Lagrange multipliers. For each subdomain, different time discretizations are used. No restrictions for relation between subdomain’s time steps are imposed. The coupling of the subdomains is forced by an acceleration continuity condition. Additionally, we use the a posteriori technique to also provide the displacement and velocity continuity at the interfaces, and hence we obtain exact continuity of all three kinematic fields. The proposed method is experimentally validated using the modified SHPB (split Hopkinson pressure bar) setup.

Solving local constraint condition problem in slave particle theory with the BRST quantization

With the Becchi–Rouet–Stora–Tyutin (BRST) quantization of gauge theory, we solve the long-standing difficult problem of the local constraint conditions, i.e. the single occupation of a slave particle per site, in the slave particle theory. This difficulty is actually caused by inconsistently dealing with the local Lagrange multiplier λ i which ensures the constraint: in the Hamiltonian formalism of the theory, λ i is time-independent and commutes with the Hamiltonian while in the Lagrangian formalism, λ i (t) becomes time-dependent and plays a role of gauge field. This implies that the redundant degrees of freedom of λ i (t) are introduced and must be removed by the additional constraint, the gauge fixing condition (GFC) ∂ t λ i (t) = 0. In literature, this GFC was missed. We add this GFC and use the BRST quantization of gauge theory for Dirac’s first-class constraints in the slave particle theory. This GFC endows λ i (t) with dynamics and leads to important physical results. As an example, we study the Hubbard model at half-filling and find that the spinon is gapped in the weak U and the system is indeed a conventional metal, which resolves the paradox that the weak coupling state is a superconductor in the previous slave boson mean field (MF) theory. For the t–J model, we find that the dynamic effect of λ i (t) substantially suppresses the d-wave pairing gap and then the superconducting critical temperature may be lowered at least a factor of one-fifth of the MF value which is of the order of 1000 K. The renormalized T c is then close to that in cuprates.

Three-field partitioned analysis of fluid–structure interaction problems with a consistent interface model

This paper proposes a new Fluid–Structure Interaction computational framework. The coupling between the solid and an incompressible fluid is formulated by means of the method of localized Lagrange multipliers (LLM). Instead of applying a direct coupling between the fluid and the structure, which is the traditional approach, LLM introduces an intermediate surface with its own degrees of freedom that is connected to the fluid and structure sides using independent fields of localized Lagrange multipliers. This approach allows the connection of non-matching meshes with mortar or classical localized methods and provides consistent dynamic equations of motion for the interface that can be integrated in parallel. Interface multipliers are later eliminated and the interface motion is used to update the fluid and structure states. This way, dedicated stand-alone software modules for the fluid and the structure are connected to a third interface system treating their interaction, thus preserving the modularity of the single-discipline software modules. Different numerical examples are solved with the proposed methodology to prove its efficiency and accuracy by running a series of classical dynamic FSI benchmark problems.

Explicit asynchronous time scheme with local push-forward stepping for discontinuous elastic wave propagation: One-dimensional heterogeneous cases and Hopkinson bar experiment

This is a presentation of robust and accurate explicit time-stepping strategy for finite element modeling of elastic discontinuous wave propagation in strongly heterogeneous, multi-material and graded one-dimensional media. One of the major issues in FEM modeling is the existence of spurious numerical stress oscillations close to theoretical wave fronts due to temporal-spatial dispersion behavior of FE discretization. The numerical strategy presented for modeling of 1D discontinuous elastic waves is based on (a) pushforward-pullback local stepping — ensuring the elimination of dispersion due to different critical time step sizes of finite elements, (b) domain decomposition via localized Lagrange multipliers — to satisfy coupling kinematics and dynamic equations , (c) asynchronous time scheme — ensuring the correct information transfer of quantities for the case of integer ratios of time step size for all domain pairs. Dispersion behaviors, existence of spurious stress oscillations, and sensitivity of the dispersion to time step size are then suppressed. The proposed method is numerically tested with regard to the rectangular step pulse elastic propagation problem considering in-space varying Young’s modulus. To prove robustness and accuracy, a comparison with results from commercial software, an analytical solution, and experimental data from partial assembly of a split Hopkinson pressure bar (SHPB) setup is provided.

A Computational Strategy for Code- and Mesh-Agnostic Nonlinear Global–Local Analysis

Global, coarse mesh and local, refined mesh modeling strategy is a common solution approach for domains spanning many orders in length scale. In this article, a nonintrusive computational strategy for iterative solution to the nonlinear behavior in the subdomains is proposed. To estimate the residual force on the interface connecting the subdomains with nonmatching discretizations, variational principles with an intermediate framework are proposed to assure satisfaction of virtual work principle in the subdomains and to assure displacement/traction compatibility at the interface. To map nodal force from one subdomain to the other, both global Lagrange multiplier (GLM) and local Lagrange multiplier (LLM) methods are discussed. Quasi-Newton iterative updates are used to correct displacements on the interface connecting the different subdomains until equilibrium is reached. A symmetric rank 1 update as well as the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update are considered for accelerating the solution convergence. The developed method is suitable for the modular construction of submodels with nonmatching discretizations and even allows coupling between subdomains analyzed using different (commercial or custom) codes. The iterative global–local coupling strategy is validated on several examples, including an L-shaped domain with local nonlinearity and a rectangular plate with a propagating crack. Another example illustrating the coupling of domains independently analyzed using commercial finite element codes ANSYS and ABAQUS is next demonstrated. Finally, the method is demonstrated to analyze semiconductor chip assemblies where plastic ratcheting of the interconnect line causes passivation coating fracture.

Coupling Soil–Fluid–Structure Domains by Localized Lagrange Multipliers Mixed Formulation (u, p) for Modeling Offshore Wind Turbine Vibration

Modeling is frequently used to design structures in a large range of engineering applications. Coupled models can be solved by several numerical procedures. This paper presents a soil–fluid–structure coupling analysis applied to offshore wind turbines design. The goal of this work is to develop a coupled structural finite element procedure using Localized Lagrange Multipliers (LLM) at idealized offshore wind turbines with poroelastic soil foundation. The poroelastic media theory of Biot is used to represent the soil domain as two-phase material which is considered to be fully saturated with water. The numerical model is validated through a fully coupled model at classical problems results. In this work, the mixed formulation (u,p) is used to model the interface frames between the domains. The interface domain behaves as fictitious porous material generating a nonsymmetrical coupling between elastic solid and potential fluid. The momentum equilibrium and mass continuity equations are solved by algebraic equation system imposed by Lagrange multipliers methodology. In order to fully model the coupled system, aerodynamic decoupling effects are implemented to separate the tower structure, the soil foundation and the rotor-nacelle assembly. This procedure is based on the blade aerodynamic characteristics. The force and moment vectors are assembled considering the aerodynamic damping coupling between inplane and outplane rotor motions. Finally, the equations of the tower-nacelle structure, ocean fluid and poroelastic soil are obtained by the classical finite element method. In addition, their interaction is modeled using LLM. The numerical model for monopile and jacked offshore wind turbine is developed in terms of time dynamic response which is evaluated for a range of physical parameters including the wind nature and soil type.

MULTISCALE MODEL WITH EMBEDDED DISCONTINUITY DISCRETE APPROXIMATION CAPABLE OF REPRESENTING FULL SET OF 3D FAILURE MODES FOR HETEROGENEOUS MATERIALS WITH NO SCALE SEPARATION

In this paper we present a 3D strongly coupled multiscale computational procedure for failure analysis of heterogeneous materials capable of representing a full set of different failure modes under various stress states. The proposed approach targets structural failure modeling by a multiscale method with no need for scale separation where a homogenized structural response is described at the macroscale, while microscale is utilized for representing a full set of 3D damage mechanisms inside the domain. The main novelty in terms of finite element method (FEM) implementation concerns macroscale elements that provide the adequate discrete approximation with embedded discontinuity for capturing localized failure with softening. Computation of element tangent stiffness matrices and residual vectors is performed by assembly of microscale elements whose contributions are statically condensed at the coarse level. A refined mesh built with corresponding microscale element discretization entirely fits in each macroscale element which acts as a container for microscale computational results. The compatibility between both scales is enforced by imposing constraint on the displacement field over the macroelement boundary producing a displacement based coupling in the spirit of localized Lagrange multipliers. Dealing with localized failure mechanisms is enabled through enlargement of such scale coupling through implementation of embedded strong discontinuity in macroscale elements. The computation is performed by using the operator split iterative solution procedure treating sequentially micro- and macroscales. Through several numerical simulations, it is demonstrated that the proposed methodology can reproduce a full set of 3D failure modes with very satisfying performance in dealing with localized failure problems.

Partitioned formulation of contact‐impact problems with stabilized contact constraints and reciprocal mass matrices

AbstractThis work presents an efficient and accuracy‐improved time explicit solution methodology for the simulation of contact‐impact problems with finite elements. The proposed solution process combines four different existent techniques. First, the contact constraints are modeled by a bipenalty contact‐impact formulation that incorporates stiffness and mass penalties preserving the stability limit of contact‐free problems for efficient explicit time integration. Second, a method of localized Lagrange multipliers is employed, which facilitates the partitioned governing equations for each substructure along with the completely localized contact penalty forces pertaining to each free substructure. Third, a method for the direct construction of sparse inverse mass matrices of the free bodies in contact is combined with the localized Lagrange multipliers approach. Finally, an element‐by‐element mass matrix scaling technique that allows the extension of the time integration step is adopted to improve the overall performance of the algorithm. A judicious synthesis of the four numerical techniques has resulted in an increased stable explicit step‐size that boosts the performance of the bipenalty method for contact problems. Classical contact‐impact numerical examples are used to demonstrate the effectiveness of the proposed methodology.

An iterative scheme of flexibility‐based component mode synthesis with higher‐order residual modal compensation

AbstractAn Iterative Flexibility‐based Component Mode Synthesis (IF‐CMS) is presented for the model reduction of partitioned structural dynamic systems via localized Lagrange multipliers. A distinct IF‐CMS feature is the inclusion of hierarchical residual flexibility at each iteration, resulting in considerably improved accuracy compared with the classical Craig–Bampton (CB) CMS method. The present IF‐CMS method does not increase the number of substructural modes during the hierarchical iteration process. In particular, to alleviate ill‐conditioning during the iteration steps, the proposed IF‐CMS method adopts the so‐called (qd, , ub)‐formulation by condensing Lagrange multipliers from the original F‐CMS method. The performance of the proposed method is evaluated through numerical examples, which illustrates improved accuracy over existing CMS methods.

A Dynamic Partitioning Method to solve the vehicle-bridge interaction problem

This paper presents a Dynamic Partitioning Method (DPM) to solve the vehicle-bridge interaction (VBI) problem via a set of exclusively second-order ordinary differential equations (ODEs). The partitioning of the coupled VBI problem follows a localized Lagrange multipliers approach that introduces auxiliary contact bodies between the vehicle’s wheels and the sustaining bridge. The introduction of contact bodies, instead of merely static points, allows the assignment of proper mass, damping and stiffness properties to the involved constrains. These properties are estimated in a systematic manner, based on a consistent application of Newton’s law of motion to mechanical systems subjected to bilateral constraints. In turn, this leads to a dynamic representation of motion constraints and associated Lagrange multipliers. Subsequently, both equations of motion and constraint equations yield a set of ODEs. This ODE formulation avoids constraint drifts and instabilities associated with differential–algebraic equations, typically adopted to solve constrained mechanical problems. Numerical applications show that, when combined with appropriate numerical analysis schemes, DPM can considerably decrease the computational cost of the analysis, especially for large vehicle-bridge systems. Thus, compared to existing methods to treat VBI, DPM is both accurate and cost-efficient.

Localized Lagrange Multipliers Mixed (u,p) Formulation Applied in Wind Turbine Analysis

Fluid–structure analysis is frequently used to design offshore structures in a large range of engineering applications. In order to install wind turbines from the seashore, it is required that its towers must be attached at the sea floor or a system needs to be developed that allows the turbine to float. Therefore, the objective of this work is to develop a coupled structural finite element analysis using localized Lagrange multipliers (LLM) at idealized monopile wind turbines. This method enables us to model large structures like wind turbine towers submerged in the ocean and determine the influence of the fluid–structure interaction on the dynamic structural response of these equipments. In this work, the mixed formulation [Formula: see text] developed takes into account the condition of irrotationality of the acoustic fluid. The classical LLM method is modified, thus, a new mixed-formulation for the interface frame has fluid pressure and solid displacement as degree of freedom. In other words, a coupling frame is introduced as fictitious porous material with the aim of non-symmetrical coupling between elastic solid and potential fluid. The solution to the non-matching meshes domain problem is also achieved with this methodology by several numerical strategies. The equations of solid and fluid domains are obtained by classical finite element method and their interaction is modeled with Localized Lagrange Multipliers. The solutions of numerical equations are obtained by direct form in a monolithic approach. The new modeling by using a porous material interface frame algorithm is verified through examples.

A new approach for nonmatching interface construction by the method of localized Lagrange multipliers

A new frame discretization method for treating non-matching discrete interfaces is presented based on the method of localized Lagrange multipliers, which introduces the frame domain lying between the two interfacing parts. The required interface compatibility conditions are then enforced independently between the master domain interface nodes and the frame nodes, and the slave domain interface nodes and the same frame nodes. The frame nodes are determined so as to satisfy the mean coordinates, including the frame node to be determined, of the nearest interface element of the master domain for each of the slave domain interface nodes. The roles of the two domains may interchange, resulting in a unique determination of the frame elements for each case. The interface compatibility conditions thus obtained satisfy energy conservation, especially when the interface gaps are unavoidable while requiring no special treatments for the boundary nodes that are often required in mortar and allied methods. Consequently, the proposed method can effectively deal with the non-matching interface regardless of the geometric complexity and element type. Numerical examples illustrate the simplicity of the proposed method and offer improved accuracy for interfaces with gaps and as good of an accuracy as other methods when there are no gaps, while preserving implementation simplicity.

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

SummaryA multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain‐to‐subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one‐dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one‐dimensional heterogeneous bar and in heterogeneous plain strain problems.

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

AbstractA variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated. Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B‐splines of high continuity and Bézier elements with a standard C0 continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.

Formulation of Flexibility-Based Component Mode Synthesis for Transient Analysis

Flexibility-based component mode synthesis (F-CMS) is one of the most popular free-interface component mode synthesis (CMS) methods. In the F-CMS method, localized Lagrange multipliers are employed along the interface boundary as a constraint condition between substructures; thus, it offers various advantages, including the perfect independence of subdomains, the easy treatment of the incompatible interface mesh, and parallelized formulation. However, the use of the Lagrange multipliers requires nonstandard specialized eigenvalue analysis procedures. This, in turn, requires a considerable modification in the standard displacement-based time integrators for transient analysis by using the F-CMS method. In the present study, the localized Lagrange multipliers are condensed by exploiting residual flexibility, and this results in a standard displacement-based reduced-order model. A report is made of the performance of the proposed method as compared to several CMS methods, with favorable features observed for the proposed method.

Virtual tetrahedral gap element to connect three-dimensional non-coincident interfaces

This study introduces a new version of a virtual tetrahedral gap element to connect partitioned structures which are independently discretized with tetrahedral elements. Tetrahedral meshes are widely used for practical engineering problems due to their simplicity. The proposed interface method employs the localized Lagrange multiplier method. The virtual tetrahedral gap elements are placed between the frame-slave and frame-master interfaces. The surface of the tetrahedral meshes is triangular; thus, a virtual tetrahedral gap element is developed. A distinct feature of the virtual tetrahedral gap element is that it has a zero-strain condition which provides the exact interface reaction forces at the non-matched interface. The proposed tetrahedral gap element handles three-dimensional interface problems more effectively than conventional segment-to-segment methods. It also provides better accuracy. The validity and robustness of the proposed method are demonstrated by several numerical examples.