Augmented Lagrangian Formulation Research Articles

Hp-Multigrid Preconditioner for a Divergence-Conforming HDG Scheme for the Incompressible Flow Problems

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 103\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^3$$\\end{document}.

Full waveform inversion and Lagrange multipliers

SUMMARY Full-waveform inversion (FWI) is an effective method for imaging subsurface properties using sparsely recorded data. It involves solving a wave propagation problem to estimate model parameters that accurately reproduce the data. Recent trends in FWI have seen a renewed interest in extended methodologies, among which source extension methods leveraging reconstructed wavefields to solve penalty or augmented Lagrangian (AL) formulations have emerged as robust algorithms, even for inaccurate initial models. Despite their demonstrated robustness on synthetic data, challenges remain, such as the lack of a clear physical interpretation and reliance on difficult-to-compute least-squares (LS) wavefields. Moreover, the literature lacks a general and through comparison of these source extension methods with each other and with the standard FWI. This paper is divided into three critical parts. In the first, a novel formulation of these methods is explored within a unified Lagrangian framework. This novel perspective permits the introduction of alternative algorithms that use LS multipliers instead of wavefields. These multiplier-oriented variants appear as regularizations of the standard FWI, are suitable to the time domain, offer tangible physical interpretations, and foster enhanced convergence efficiency. The second part of the paper delves into understanding the underlying mechanisms of these techniques. This is achieved by solving the associated non-linear equations using iterative linearization and inverse scattering methods. The paper provides insight into the role and significance of Lagrange multipliers in enhancing the linearization of the equations. It explains how different methods estimate multipliers or make approximations to increase computing efficiency. Additionally, it presents a new physical understanding of the Lagrange multiplier used in the AL method, highlighting how important it is for improving algorithm performance when compared to penalty methods. In the final section, the paper presents numerical examples that compare different methods within a unified iterative algorithm, utilizing benchmark Marmousi and 2004 BP salt models.

Rate-independent gradient-enhanced crystal plasticity theory — Robust algorithmic formulations based on incremental energy minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes.

Numerically efficient and robust interior-point algorithm for finite strain rate-independent crystal plasticity

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation.

Augmented Lagrangian index-3 semi-recursive formulations with projections

Sensitivity analysis represents a powerful tool for the optimization of multibody system dynamics. The performance of a gradient-based optimization algorithm is strongly tied to the dynamic and the sensitivity formulations considered. The accuracy and efficiency are critical to any optimization problem, thus they are key factors in the selection of the dynamic and sensitivity analysis approaches used to compute an objective function gradient. Semi-recursive methods usually outperform global methods in terms of computational time, even though they involve sometimes demanding recursive procedures. Semi-recursive methods are well suited to be combined with different constraints enforcement schemes as the augmented Lagrangian index-3 formulation with velocity and acceleration projections (ALI3-P), taking advantage of the robustness, accurate fulfillment of constraint equations and the low computational burden. The sensitivity analysis of the semi-recursive ALI3-P formulation is studied in this document by means of the direct differentiation method. As a result, a semi-recursive ALI3-P sensitivity formulation is developed for an arbitrary reference point selection, and then two particular versions are unfolded and implemented in the general purpose multibody library MBSLIM, using as reference point the center of mass (RTdyn0) or the global origin of coordinates (RTdyn1). Besides, the detailed derivatives of the recursive terms are provided, which will be useful not only for the direct sensitivity formulation presented herein, but also for other sensitivity formulations relying on the same recursive expressions. The implementation has been tested in two numerical experiments, a five-bar benchmark problem and a buggy vehicle.

Multiplier waveform inversion: A reduced-space full-waveform inversion by the method of multipliers

Full-waveform inversion (FWI) is a nonlinear optimization problem that addresses the estimation of subsurface model parameters by matching the predicted to the observed seismograms. We formulate FWI as a constrained optimization problem in which the regularization term is minimized subject to the nonlinear data matching constraints. Unlike standard FWI, which solves this regularized problem with a penalty method, we use an augmented Lagrangian formulation in the framework of the method of multipliers (MM). This leads to a two-step recursive algorithm, which is called multiplier waveform inversion (MWI). First, the primal step solves an unconstrained minimization problem, which is similar to the standard FWI but with a modified data term. These modified data are obtained by adding the Lagrange multipliers to the data residuals of the classical FWI. Then, the dual step updates the Lagrange multipliers with basic gradient ascent steps. In this framework, they reduce to the running sum of the data residuals of the previous iterations. The performance of the overall algorithm is improved by considering that the MM does not require the exact solution of the primal step at each iteration. In fact, convergence is attained when the primal subproblem is solved with one (without an inner loop) iteration of a gradient-based method at each iteration of the MWI. The new algorithm can be easily implemented in existing FWI codes without computational overhead by adding the Lagrange multipliers to the data residuals at each iteration. We find with numerical examples that MWI has a faster convergence speed and much improved stability compared with FWI and can converge to an accurate solution of the inverse problem in the absence of low-frequency data, even with a constant step size.

A shape optimization algorithm based on directional derivatives for three‐dimensional contact problems

AbstractThis work deals with shape optimization for contact mechanics. More specifically, the linear elasticity model is considered under the small deformations hypothesis, and the elastic body is assumed to be in contact (sliding or with Tresca friction) with a rigid foundation. The mathematical formulations studied are two regularized versions of the original variational inequality: the penalty formulation and the augmented Lagrangian formulation. In order to get the shape derivatives associated to those two non‐differentiable formulations, we follow an approach based on directional derivatives introduced in previous works. This allows us to develop a gradient‐based topology optimization algorithm, built on these derivatives and a level‐set representation of shapes. The algorithm also benefits from a mesh‐cutting technique, which gives an explicit representation of the shape at each iteration, and enables us to apply the boundary conditions strongly on the contact zone. The different steps of the method are detailed. Then, to validate the approach, some numerical results on two‐dimensional and three‐dimensional benchmarks are presented.

Optimal structures for failure resistance under impact

The complex physics and numerous failure modes of structural impact creates challenges when designing for impact resistance. While simple geometries of layered material are conventional, advances in 3D printing and additive manufacturing techniques have now made tailored geometries or integrated multi-material structures achievable. Here, we apply gradient-based topology optimization to the design of such structures. We start by constructing a variational model of an elastic–plastic material enriched with gradient phase-field damage, and present a novel method to efficiently compute its transient dynamic time evolution. We consider a finite element discretization with explicit updates for the displacements. The damage field is solved through an augmented Lagrangian formulation, splitting the operator coupling between the nonlinearity and non-locality. Sensitivities over this trajectory are computed through the adjoint method, and we develop a numerical method to solve the resulting adjoint dynamical system. We demonstrate this formulation by studying the optimal design of 2D solid–void structures undergoing blast loading. Then, we explore the trade-offs between strength and toughness in the design of a spall-resistant structure composed of two materials of differing properties undergoing dynamic impact.

Robust Multigrid Techniques for Augmented Lagrangian Preconditioning of Incompressible Stokes Equations with Extreme Viscosity Variations

We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear viscous flow problems, and they can have severely inhomogeneous and anisotropic coefficients. Using an augmented Lagrangian formulation for the incompressibility constraint makes the Schur complement easier to approximate, but results in a nearly singular (1,1)-block in the Stokes system. We present eigenvalue estimates for the quality of the Schur complement approximation. To cope with the near-singularity of the (1,1)-block, we extend a multigrid scheme with a discretization-dependent smoother and transfer operators from triangular/tetrahedral to the quadrilateral/hexahedral finite element discretizations $[\mathbb{Q}_k]^d\times \mathbb{P}_{k-1}^{\text{disc}}$, $k\geq 2$, $d=2,3$. Using numerical examples with scalar and with anisotropic fourth-order tensor viscosity arising from linearization of a viscoplastic constitutive relation, we confirm the robustness of the multigrid scheme and the overall efficiency of the solver. We present scalability results using up to 28,672 parallel tasks for problems with up to 1.6 billion unknowns and a viscosity contrast up to ten orders of magnitude.

3D thermoelastic solids under non‐linear interface thermal and orthotropic frictional contact conditions

AbstractThis article presents a robust and efficient methodology to study three‐dimensional thermoelastic contact problems under orthotropic friction conditions, including the effects of non‐linear thermal contact conductance and convective boundary conditions at the interstitial contact zone. The proposed methodology considers the boundary element method to compute the thermomechanical influence coefficients and an augmented Lagrangian formulation to ensure the fulfillment of all these thermomechanical contact conditions. The resulting non‐linear equation set have been solved by an efficient proposed iterative Uzawa scheme. The proposed formulation has been validated by comparison with some available results in the literature and later on, it is considered to study how the thermoelastic contact variables are affected by both: the non‐linear interface thermal and the orthotropic friction conditions in different engineering problems such as a brake disc‐pad contact system.

A regularization method for solving dynamic problems with singular configuration

In the simulation process for multi-body systems, the generated redundant constraints will result in ill-conditioned dynamic equations, which are not good for stable simulations when the system motion proceeds near a singular configuration. In order to overcome the singularity problems, the paper presents a regularization method with an explicit expression based on Gauss principle, which does not need to eliminate the constraint violation after each iteration step compared with the traditional methods. Then the effectiveness and stability are demonstrated through two numerical examples, a slider-crank mechanism and a planar four-bar linkage. Simulation results obtained with the proposed method are analyzed and compared with augmented Lagrangian formulation and the null space formulation in terms of constraints violation, drift mechanical energy and computational efficiency, which shows that the proposed method is suitable to perform efficient and stable dynamic simulations for multi-body systems with singular configurations.

The eccentric disk and its eccentric behavior

Providing example problems that stimulate curiosity and cultivate intuition among students is a prominent responsibility for physics educators. A concept that can be particularly challenging for students to grasp and educators to convey is the relationship between frictional and normal forces in rolling motion. A system that may be a useful tool to exhibit a rich interplay between these forces is a rolling eccentric disk. An eccentric disk has a non-constant normal force and therefore has four distinct phases of motion: oscillations about a stable equilibrium, roll without slip, roll with slip, and hop. Thus, this paper examines the eccentric disk rolling down an incline as a valuable teaching medium and explores its peculiar behavior. The rolling system is analytically modeled using an augmented Lagrangian formulation, solved with numerical integration, and experimentally realized. Then, the results are presented and discussed in detail.

On a flexible multibody modelling approach using FE-based contact formulation for describing gear transmission error

This paper provides an insight into the efficiency and accuracy of a multibody approach to model gear transmission error. The multibody model is based on an augmented Lagrangian contact formulation considering a surface-to-surface contact detection. The case of spur and helical gears transmitting power between parallel shafts is considered. The static transmission error is computed without any assumptions about the contact lines positions and orientations. Tooth and wheel body flexibility and tooth profile deviations are taken into account. The main objective of this study is to evaluate the efficiency and possibilities of the proposed methodology. To this end, the static transmission error is benchmarked against the results obtained from a classical approach which is based on the computation of the equation describing the static equilibrium of the gear pair for a set of successive positions of the driving wheel.

Alternating optimization of design and stress for stress-constrained topology optimization

Handling stress constraints is an important topic in topology optimization. In this paper, we introduce an interpretation of stresses as optimization variables, leading to an augmented Lagrangian formulation. This formulation takes two sets of optimization variables, i.e., an auxiliary stress variable per element, in addition to a density variable as in conventional density-based approaches. The auxiliary stress is related to the actual stress (i.e., computed by its definition) by an equality constraint. When the equality constraint is strictly satisfied, an upper bound imposed on the auxiliary stress design variable equivalently applies to the actual stress. The equality constraint is incorporated into the objective function as linear and quadratic terms using an augmented Lagrangian form. We further show that this formulation is separable regarding its two sets of variables. This gives rise to an efficient augmented Lagrangian solver known as the alternating direction method of multipliers (ADMM). In each iteration, the density variables, auxiliary stress variables, and Lagrange multipliers are alternatingly updated. The introduction of auxiliary stress variables enlarges the search space. We demonstrate the effectiveness and efficiency of the proposed formulation and solution strategy using simple truss examples and a dozen of continuum structure optimization settings.

Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics

In this paper, we propose in a Hilbertian setting a second-order time-continuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are, respectively, associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.

Optimized design of a tungsten–copper functionally graded material monoblock for minimal von Mises stress meeting the material operational temperature window

Functionally graded materials (FGMs) are a means to remove discrete material interfaces which lead to high local stress concentrations, such as the tungsten–copper (W–Cu) interface of the current ITER monoblock design. This paper employs adjoint-based optimization methods to identify the highest potential reduction of stresses that could be reached with these materials, while ensuring that the local temperature does not exceed the material temperature operational window. The cheap sensitivity evaluation inherent to the adjoint approach enables the optimization of the detailed 3D material distribution. Furthermore, a novel optimization method based on an augmented Lagrangian formulation is proposed that allows accurate treatment of the material temperature window constraints. The temperature and stresses are modelled by the steady heat conduction and Navier’s equation, respectively. We compare the results of different optimization formulations, with cost functions based on the von Mises stress and corresponding yield criterion and considering different values of the stress free temperature. To assess the performance under off-design conditions, two optimized designs were chosen and compared to the ITER and flat tile (FT) design, which consists of a copper block protected by a tungsten layer on top. The optimized designs lead to a factor 2–4 decrease in maximal stress near the original W–Cu interface of the FT design and a factor 10 decrease in yield criterion measure near the cooling duct. Under off-design conditions, they realized a factor 2–10 decrease in yield criterion in the upper part of the monoblock. This confirms numerically that FGMs can lead to significant design improvements. Finally, the inclusion of the material temperature operation window constraints leads to a decrease of 30–55 vol% W compared to the unconstrained cases, thus profoundly influencing the final design. The stress free temperature was found to have a comparably weaker influence on the final design with differences of 5–30 vol% W.

A time-stepping method for multibody systems with frictional impacts based on a return map and boundary layer theory

This work presents a new numerical integration method for determining dynamics of a class of multibody systems involving impact and friction. Specifically, these systems are subject to a set of equality constraints and can exhibit single frictional impact events. Such events are associated to significant numerical stiffness, appearing in the equations of motion. The new method is a time-stepping scheme, involving proper incorporation of a novel return mapping into an augmented Lagrangian formulation, developed recently for systems with bilateral constraints only. Namely, when an impact is detected during a time step, this map is applied at the end of the step in order to bring the system position back to the configuration manifold with the allowable motions. The construction of this map is based on the concept of Jacobi fields on non-flat manifolds. Moreover, once an impact event is detected, the post-impact state is determined by employing a combination of analytical and numerical tools. First, a proper coordinate transformation is performed, bringing the system into a new set of coordinates, which are suitable for describing the impact dynamics. In these coordinates, the dominant dynamics is described by a system of three equations of motion only, which are valid during the short contact interval. In addition, these equations are geometrically discretized by using appropriate cubic splines on the configuration manifold. In this way, the inherent numerical stiffness of the class of systems examined is properly addressed, since it is restricted to a space with a much smaller dimension and a much shorter time scale. Finally, the accuracy and efficiency of the new method is demonstrated by applying it to a selected set of mechanical examples.

ElfPlace: Electrostatics-Based Placement for Large-Scale Heterogeneous FPGAs

<monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">elfPlace</monospace> is a flat nonlinear placement algorithm for large-scale heterogeneous field-programmable gate arrays (FPGAs). We adopt the analogy between placement and electrostatic systems initially proposed by <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ePlace</monospace> and extend it to tackle heterogeneous blocks in FPGA designs. To achieve satisfiable solution quality with fast and robust numerical convergence, an augmented Lagrangian formulation together with a preconditioning technique and a normalized subgradient-based multiplier updating scheme are proposed. Besides pure-wirelength minimization, we also propose a unified instance area adjustment scheme to simultaneously optimize routability, pin density, and downstream clustering compatibility. We further propose run-to-run deterministic GPU acceleration techniques to speedup the global placement. Our experiments on the ISPD 2016 benchmark suite show that <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">elfPlace</monospace> outperforms four state-of-the-art FPGA placers <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">UTPlaceF</monospace> , <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RippleFPGA</monospace> , <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GPlace3.0</monospace> , and <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">UTPlaceF-DL</monospace> by 13.5%, 10.2%, 8.8%, and 7.0%, respectively, in routed wirelength with competitive runtime.

Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Because of the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

Augmented Lagrangian index-3 semi-recursive formulations with projections

MBSLIM (Multibody Systems at Laboratorio de Ingenieria Mecanica) multibody library includes some global formulations for the dynamics and sensitivity analysis of multibody systems. The extension of the library to accommodate topological formulations in relative (joint) coordinates and their implementation are going to be described in two separate works, this one being devoted to dynamics and the second one to a sensitivity analysis. With this extension in the scope, some topological semi-recursive formulations derived in the past are revisited, generalized and reformulated. The need for generalization of the previously published formulations was detected because the equations proposed were not general enough to be integrated in an all-purpose multibody library in natural coordinates like MBSLIM, especially because both set of coordinates need to coexist, the definition of the mechanisms has to be the original one and the library has to be automatic and all the existing models have to work with the new approach. Moreover the new solver takes advantage of some problems solved in natural coordinates, like the initial position and initial velocity problems for closed-loop systems. Finally, to test the new equations, two benchmark problems are presented and their results compared: a spatial slider–crank mechanism and a buggy vehicle model.