Multiplier Field Research Articles

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

Rate-independent dissipation in phase-field modelling of displacive transformations

In this paper, rate-independent dissipation is introduced into the phase-field framework for modelling of displacive transformations, such as martensitic phase transformation and twinning. The finite-strain phase-field model developed recently by the present authors is here extended beyond the limitations of purely viscous dissipation. The variational formulation, in which the evolution problem is formulated as a constrained minimization problem for a global rate-potential, is enhanced by including a mixed-type dissipation potential that combines viscous and rate-independent contributions. Effective computational treatment of the resulting incremental problem of non-smooth optimization is developed by employing the augmented Lagrangian method. It is demonstrated that a single Lagrange multiplier field suffices to handle the dissipation potential vertex and simultaneously to enforce physical constraints on the order parameter. In this way, the initially non-smooth problem of evolution is converted into a smooth stationarity problem. The model is implemented in a finite-element code and applied to solve two- and three-dimensional boundary value problems representative for shape memory alloys.

Solvation in atomic liquids: connection between Gaussian field theory and density functional theory

For the problem of molecular solvation, formulated as a liquid submitted to the external potential field created by a molecular solute of arbitrary shape dissolved in that solvent, we draw a connection between the Gaussian field theory derived by David Chandler [Phys. Rev. E, 1993, 48, 2898] and classical density functional theory. We show that Chandler's results concerning the solvation of a hard core of arbitrary shape can be recovered by either minimising a linearised HNC functional using an auxiliary Lagrange multiplier field to impose a vanishing density inside the core, or by minimising this functional directly outside the core --- indeed a simpler procedure. Those equivalent approaches are compared to two other variants of DFT, either in the HNC, or partially linearised HNC approximation, for the solvation of a Lennard-Jones solute of increasing size in a Lennard-Jones solvent. Compared to Monte-Carlo simulations, all those theories give acceptable results for the inhomogeneous solvent structure, but are completely out-of-range for the solvation free-energies. This can be fixed in DFT by adding a hard-sphere bridge correction to the HNC functional.

Interfacial stabilization at finite strains for weak and strong discontinuities in multi-constituent materials

A stabilized interface formulation is developed for debonding at finite strains across general bimaterial interfaces by embedding stabilized Discontinuous Galerkin (DG) ideas in the Continuous Galerkin method. Introducing an interfacial gap function that evolves subject to constraints imposed by opening and/or sliding interfaces, the proposed Variational Multiscale DG (VMDG) method seamlessly tracks interface debonding by treating damage and friction in a unified way. An internal variable formalism together with the notion of irreversibility of damage results in a set of evolution equations for the gap function. Evolution of the debonding surfaces requires interfacial stabilization that is developed based on residual-based stabilization concepts. Tension debonding, compression damage, and frictional sliding are accommodated, and return mapping algorithms in the presence of evolving strong discontinuities are developed. A significant contribution of the paper is the consistently derived method to model the Lagrange multiplier field via interfacial flux and jump terms and variational embedding of various nonlinear interfacial debonding models at the interfacial boundaries. This derivation variationally embeds the interfacial kinematic models that are crucial to capturing the physical and mathematical properties involving large stains and damage. A set of representative test cases highlights the salient features of the proposed VMDG method and confirm its robustness and range of applicability.

Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid–thin structure interaction analysis, with application to heart valve modeling

This paper discusses a method of stabilizing Lagrange multiplier fields used to couple thin immersed shell structures and surrounding fluids. The method retains essential conservation properties by stabilizing only the portion of the constraint orthogonal to a coarse multiplier space. This stabilization can easily be applied within iterative methods or semi-implicit time integrators that avoid directly solving a saddle point problem for the Lagrange multiplier field. Heart valve simulations demonstrate applicability of the proposed method to 3D unsteady simulations. An appendix sketches the relation between the proposed method and a high-order-accurate approach for simpler model problems.

A discussion on the mechanics of lipid membranes: Lagrange multipliers and a singular substrate

We present rigorous analysis of Lagrange multipliers arising in the variational computations of a membrane’s energy functional. Through the convergence analysis, it is found that the local Lagrange multiplier is sensitive to the properties of membranes. Unlike the local Lagrange multiplier, the bulk liquid incompressibility condition via the global Lagrangian field acts as an additional boundary input rather than a constraint, in the perspective of the surrounding membrane. With the enhanced understanding of Lagrange multiplier fields, a membrane–substrate interaction problem is considered where singularities are present at the cross section of the substrate. A viable analytical solution is obtained under the modified boundary conditions. In addition, a complete analysis of necessary boundary condition in the case of flat surfaces is presented within the prescription of superposed incremental deformations.

Edge stabilization and consistent tying of constituents at Neumann boundaries in multi‐constituent mixture models

SummaryA mixture‐theory‐based model for multi‐constituent solids is presented where each constituent is governed by its own balance laws and constitutive equations. Interactive forces between constituents that emanate from maximization of entropy production inequality provide the coupling between constituent‐specific balance laws and constitutive models. The deformation of multi‐constituent mixtures at the Neumann boundaries requires imposing inter‐constituent coupling constraints such that the constituents deform in a self‐consistent fashion. A set of boundary conditions is presented that accounts for the non‐zero applied tractions, and a variationally consistent method is developed to enforce inter‐constituent constraints at Neumann boundaries in the finite deformation context. The new method finds roots in a local multiscale decomposition of the deformation map at the Neumann boundary. Locally satisfying the Lagrange multiplier field and subsequent modeling of the fine scales via edge bubble functions result in closed‐form expressions for a generalized penalty tensor and a weighted numerical flux that are free from tunable parameters. The key novelty is that the consistently derived constituent coupling parameters evolve with material and geometric nonlinearity, thereby resulting in optimal enforcement of inter‐constituent constraints. Various benchmark problems are presented to validate the method and show its range of application. Copyright © 2016 John Wiley & Sons, Ltd.

A frictional contact algorithm for implicit material point method

The explicit material point method (MPM) works successfully in modeling high frequency problems, but it is very computationally expensive in simulating low frequency with small time steps or quasi-static problems. Thus, several groups have developed an implicit MPM for modeling low frequency problems. Recently, a few attempts were undertaken to investigate the contact problems using the implicit MPM but the accuracy was dissatisfactory. In this paper, an augmented Lagrange formulation for the frictional inequality constraints is introduced. A discretization of the Lagrange multiplier field based on the background grid is proposed to establish the implicit MPM framework with the contact algorithm. To reduce the complexity of the solution, the Uzawa algorithm is employed to decouple the unknown variables and the Lagrange multipliers. Finally, the resulting sequent nonlinear equations are solved by the Newton method, in which the tangential matrix is assembled explicitly. By using the compressed sparse row (CSR) technique, the total storage of the matrix can be greatly reduced. Numerical studies show that the computational efficiency and accuracy of the implicit MPM with the proposed contact algorithm are much higher than the explicit MPM.

Topological induced gravity

We propose a topological model of induced gravity (pregeometry) where both Newton’s coupling constant and the cosmological constant appear as integration constants in solving field equations. The matter sector of a scalar field is also considered, and by solving field equations it is shown that various types of cosmological solutions in the Friedmann–Robertson–Walker (FRW) universe can be obtained. A detailed analysis is given of the meaning of the BRST transformations, which make the induced gravity be a topological field theory, by means of the canonical quantization analysis, and the physical reason why such BRST transformations are needed in the present formalism is clarified. Finally, we propose a dynamical mechanism for fixing the Lagrange multiplier fields by following the Higgs mechanism. The present study clearly indicates that the induced gravity can be constructed at the classical level without recourse to quantum fluctuations of matter and suggests an interesting relationship between the induced gravity and the topological quantum-field theory (TQFT).

The domain interface method in non-conforming domain decomposition multifield problems

The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya–Babuška–Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.

A weighted point-based formulation for isogeometric contact

In the context of isogeometric contact analysis, surfaces of objects can be described smoothly due to the high continuity of the involved shape functions. This facilitates construction of a continuous normal field which may be beneficial for modeling of contact problems. This property allows for stable collocation of the virtual contact work using considerable less evaluation points than numerical integration, while the approximation quality of the computational results is almost the same. In this context, a weighted point-based method (PTS+) is introduced as an extension of the Point-To-Segment method (PTS) proposed previously by Matzen et al. (2013). The weights enable transmission of contact stresses instead of contact forces along the contact surfaces. In normal direction the non-penetration condition is enforced by the Lagrange multiplier method, combined with the penalty method in tangential direction. Corresponding to the collocation weights, a constant or NURBS Lagrange multiplier field is introduced. A comparison between the collocation-based contact formulations PTS, PTS+ and a mortar method is drawn, showing that weighted collocation is able to achieve almost the same accuracy as the integration-based mortar formulation.

Instanton calculus of Lifshitz tails

For noninteracting particles moving in a Gaussian random potential, there exists a disagreement in the literature on the asymptotic expression for the density of states in the tail of the band. We resolve this discrepancy. Further we illuminate the physical facet of instantons appearing in replica and supersymmetric derivations with another derivation employing a Lagrange multiplier field.

A stabilized, symmetric Nitsche method for spatially localized plasticity

A heterogeneous interface method is developed for combining primal displacement and mixed displacement-pressure formulations across nonconforming finite element meshes to treat volume-preserving plastic flow. When the zone of inelastic response is localized within a larger domain, significant computational savings can be achieved by confining the mixed formulation solely to the localized region. The method's distinguishing feature is that the coupling terms for joining dissimilar element types are derived from a time-discrete free energy functional, which is based on a Lagrange multiplier formulation of the interface constraints. Incorporating residual-based stabilizing terms at the interface enables the condensation of the multiplier field, leading to a symmetric Nitsche formulation in which the interface operators respect the differing character of the governing equations in each region. In a series of numerical problems, the heterogeneous interface method achieved comparable results on coarser meshes as those obtained from applying the mixed formulation throughout the domain.

Thermo-elastic extended meshfree method for fracture without crack tip enrichment

This is the first manuscript presenting an extended meshfree method for thermo- elastic fracture which does not exploit a crack tip enrichment. The crack is modeled by partition of unity enrichment of the displacement and temperature field. Only a step function is employed that facilitates the implementation. To ensure that crack tip is at the correct position, a Lagrange multiplier field ahead of the crack tip is introduced along a line. The Lagrange multiplier nodal parameters are discretised with the available meshfree functions. Two benchmark examples illustrate the efficiency of the method.

Singular elastostatic fields near the notch vertex of a Mooney–Rivlin hyperelastic body

Within the framework of fully nonlinear fracture mechanics under plane deformation conditions, boundary value problem equations for the V-notch vertex problem in an incompressible Mooney–Rivlin material are deduced. Using an asymptotic procedure, the deformation, the Lagrange multiplier and stress fields near the notch vertex are computed and their principal properties are illustrated. First, it is shown that the singularity order depend on the V-notch angle. Second, a change of the singular term for a critical angle was revealed and a logarithmic singular term appears. Based on the results obtained, the deformed V-notch vertex lips near the tip was drawn. The V-notch vertex is shown to open, but not necessary in a symmetric manner. Analysis of the Cauchy stress tensor shows that the component σ 22 dominates the stress field. Some discrepancy with the linear theory have emerged.

Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

AbstractThe purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

The two faces of mimetic Horndeski gravity: disformal transformations and Lagrange multiplier

We show that very general scalar-tensor theories of gravity (including, e.g., Horndeski models) are generically invariant under disformal transformations. However there is a special subset, when the transformation is not invertible, that yields new equations of motion which are a generalization of the so-called “mimetic” dark matter theory recently introduced by Chamsedinne and Mukhanov. These conclusions hold true irrespective of whether the scalar field in the action of the assumed scalar-tensor theory of gravity is the same or different than the scalar field involved in the transformation. The new equations of motion for our general mimetic theory can also be derived from an action containing an additional Lagrange multiplier field. The general mimetic scalar-tensor theory has the same number of derivatives in the equations of motion as the original scalar-tensor theory. As an application we show that the simplest mimetic scalar-tensor model is able to mimic the cosmological background of a flat FLRW model with a barotropic perfect fluid with any constant equation of state.

An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves

In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm.The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance.To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

In situ Underwater Measurements of Red Coral: Non-Intrusive Approach Based on Coded Targets and Photogrammetry

Since a decade, we develop a photogrammetry tool dedicated to monitoring red coral populations. This tool is used to collect, effectively, accurate data for the study of population dynamics of red coral. In this article, we present new developments that automate the steps of calculating scale and orientation of images. This automation is made possible through the use of coded targets. In measuring red coral by random quadrat sampling, scuba-divers use a set of photographs on which are visible colonies surrounded by a 20×20 cm quadrat. The analysis of the photographs provides reliable measurements on colony sizes (basal diameter and maximum height), occurrence of breakage of colonies and the occurrence of necrosis. To minimize the divers' tasks during the acquisition phase, we opted for multi-view photogrammetric acquisition using a single device to easily adapt the measurement procedure to the scene configuration. The material is quite light: one digital camera, two electronic strobes and a simple procedure with a few photographs taken for each site. To facilitate the measurement phase of colony sizes; the exploitation of photographs consists of four key steps: orientation, scaling, measurement of the characteristic points of coral colonies and result validation (checking measurement consistency to detect possible errors in measurement or interpretation). Since the context of the shooting can vary widely, dominant colors, contrast, etc. may often change. In order to have a stable and common reference in all photographs independently of the site, we decided to use the quadrat in the scene for orientation and scaling phases. The automation of orientation and the lack of constraints to adapt the analytical technique to the features of each site offer the possibility to multiply field surveys and to measure hundreds of quadrats from several different populations in a very efficient manner. The measurement results are exported into a spreadsheet application and integrated into the biologists' workflow. The results obtained from different red coral populations displaying contrasting characteristics (small versus large colony sizes) are presented and discussed at the end of this article.

AUTOMATING THE MEASUREMENT OF RED CORAL IN SITU USING UNDERWATER PHOTOGRAMMETRY AND CODED TARGETS

Abstract. A photogrammetry tool dedicated to the monitoring of red coral populations in situ has been developed by LSIS in Marseille (France). This tool is used to collect in an efficient and precise manner key data for the study of the population dynamics of red coral. In selected red coral populations, scuba-divers obtain a series of photographs from the permanent plots (about 2 m2) on an annual basis. To facilitate the photographic sampling and measurements, the scuba-divers use a 20 x 20 cm quadrat to cover the permanent plots. The analysis of the photographs provides reliable measurements on colony sizes (basal diameter and maximum height), occurrence of breakage of colonies and the occurrence of necrosis. To minimize the divers' tasks during the acquisition phase, we opted for stereoscopic acquisition using a single device to easily adapt the measurement procedure to the scene configuration. The material is quite light, one camera and two electronic strobes and a simple procedure with two photographs taken for each site. To facilitate the measurement phase of colony sizes; the exploitation of photographs consists of four key steps: orientation, scaling, measurement of the characteristic points of coral colonies and result validation (checking measurement consistency to detect possible errors in measurement or interpretation). Since the context of the shooting can vary widely, dominant colors, contrast, etc. may often change. In order to have a stable and common reference in all photographs independently of the site, we decided to always include a quadrat in the scene which then will be used for the orientation and scaling phases. The automation of orientation and the lack of constraints to adapt the analytical technique to the features of each site offer the possibility to multiply field surveys and to measure hundreds of quadrats from several different populations in a very efficient manner. The measurement results are exported into a spreadsheet application and integrated into the biologists' workflow. The results obtained from different red coral populations displaying contrasting characteristics (small versus large colony sizes) are presented and discussed at the end of this article.