Showing papers in "International Journal for Numerical Methods in Engineering in 2015"

CutFEM: Discretizing geometry and partial differential equations

Abstract: We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations

Abstract: In this work we present a stable proper orthogonal decomposition (POD)-Galerkin approximation for parametrized steady incompressible Navier-Stokes equations with low Reynolds number. Supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced-order system, considering in particular the fulfillment of an inf-sup condition. The stability analysis is first carried out from a theoretical standpoint, then confirmed by numerical tests performed on a parametrized two-dimensional backward facing step flow.

Computational homogenization of nonlinear elastic materials using neural networks

Abstract: In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using Neural Networks (NN). In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by FEM through random sampling in the parameter space, and NN are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider 3D Representative Volume Elements (RVE) and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two-scale example of heterogeneous structure with graded nonlinear properties.

Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models

Abstract: Summary The computational efficiency of a typical, projection-based, nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the scalar projections onto a subspace of a residual vector. For implicit computations, it also hinges on the additional efficient approximation of similar projections of the Jacobian of this residual with respect to the solution. The computation of both approximations is often referred to in the literature as hyper reduction. To this effect, this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced-order models of solids and structures of the recently developed energy-conserving mesh sampling and weighting (ECSW) hyper reduction method. Unlike most alternative approaches, this method approximates the scalar projections of residuals and/or Jacobians directly, instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest. In this paper, it is shown that ECSW distinguishes itself furthermore from other hyper reduction methods through its preservation of the Lagrangian structure associated with Hamilton's principle. For second-order dynamical systems, this enables it to also preserve the numerical stability properties of the discrete system to which it is applied. It is also shown that for a fixed set of parameter values, the approximation error committed online by ECSW is bounded by its counterpart error committed off-line during the training of this method. Therefore, this error can be estimated in this case a priori and controlled. The performance of ECSW is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems. For both of them, ECSW is shown to preserve numerical stability and deliver the desired level of accuracy, whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability. The potential of ECSW for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme. For this simulation, ECSW is reported to enable the reduction of the CPU time required by the high-dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude, while achieving a very good level of accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

Abstract: Crack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase-field model for strongly anisotropic fracture, which resorts to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations.

Data-driven model reduction for the Bayesian solution of inverse problems

Abstract: United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Applied Mathematics Program Award DE-FG02-08ER2585)

A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics

Abstract: Summary We present a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; weak greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M3) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract, even from these results with real data, the numerical trends indicated by the theoretical convergence and stability analyses. Copyright © 2014 John Wiley & Sons, Ltd.

Novel structural modeling and mesh moving techniques for advanced fluid-structure interaction simulation of wind turbines

Abstract: Summary In this paper, we target more advanced fluid–structure interaction (FSI) simulations of wind turbines than reported previously. For this, we illustrate how the recent advances in isogeometric analysis of thin structures may be used for efficient structural mechanics modeling of full wind turbine structures, including tower, nacelle, and blades. We consider both horizontal axis and vertical axis wind turbine designs. We enhance the sliding–interface formulation of aerodynamics, previously developed to handle flows about mechanical components in relative motion such as rotor–tower interaction to allow nonstationary sliding interfaces. To accommodate the nonstationary sliding interfaces, we propose a new mesh moving technique and present its mathematical formulation. The numerical examples include structural mechanics verification for the new offshore wind turbine blade design, FSI simulation of a horizontal axis wind turbine undergoing yawing motion as it turns into the wind and FSI simulation of a vertical axis wind turbine. The FSI simulations are performed at full scale and using realistic wind conditions and rotor speeds. Copyright © 2014 John Wiley & Sons, Ltd.

Progressive construction of a parametric reduced-order model for PDE-constrained optimization

Abstract: Summary An adaptive approach to using reduced-order models (ROMs) as surrogates in partial differential equations (PDE)-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given ROM is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the high-dimensional model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a reduced-order basis is also presented, along with a methodology for computing sensitivities of the ROM that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared with the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%. Copyright © 2014 John Wiley & Sons, Ltd.

General topology optimization method with continuous and discrete orientation design using isoparametric projection

Abstract: A general topology optimization method, which is capable of simultaneous design of density and orientation of anisotropic material, is proposed by introducing orientation design variables in addition to the density design variable. In this work, the Cartesian components of the orientation vector are utilized as the orientation design variables. The proposed method supports continuous orientation design, which is out of the scope of discrete material optimization approaches, as well as design using discrete angle sets. The advantage of this approach is that vector element representation is less likely to fail into local optima because it depends less on designs of former steps, especially compared with using the angle as a design variable (Continuous Fiber Angle Optimization) by providing a flexible path from one angle to another with relaxation of orientation design space. An additional advantage is that it is compatible with various projection or filtering methods such as sensitivity filters and density filters because it is free from unphysical bound or discontinuity such as the one at theta = 2 pi and theta = 0 seen with direct angle representation. One complication of Cartesian component representation is the point-wise quadratic bound of the design variables; that is, each pair of element values has to reside in a given circular bound. To overcome this issue, we propose an isoparametric projection method, which transforms box bounds into circular bounds by a coordinate transformation with isoparametric shape functions without having the singular point that is seen at the origin with polar coordinate representation. A new topology optimization method is built by taking advantage of the aforementioned features and modern topology optimization techniques. Several numerical examples are provided to demonstrate its capability. Copyright (C) 2014 John Wiley & Sons, Ltd.

Multidimensional parallelepiped model—a new type of non-probabilistic convex model for structural uncertainty analysis

Abstract: Summary Non-probabilistic convex models need to be provided only the changing boundary of parameters rather than their exact probability distributions; thus, such models can be applied to uncertainty analysis of complex structures when experimental information is lacking. The interval and the ellipsoidal models are the two most commonly used modeling methods in the field of non-probabilistic convex modeling. However, the former can only deal with independent variables, while the latter can only deal with dependent variables. This paper presents a more general non-probabilistic convex model, the multidimensional parallelepiped model. This model can include the independent and dependent uncertain variables in a unified framework and can effectively deal with complex ‘multi-source uncertainty’ problems in which dependent variables and independent variables coexist. For any two parameters, the concepts of the correlation angle and the correlation coefficient are defined. Through the marginal intervals of all the parameters and also their correlation coefficients, a multidimensional parallelepiped can easily be built as the uncertainty domain for parameters. Through the introduction of affine coordinates, the parallelepiped model in the original parameter space is converted to an interval model in the affine space, thus greatly facilitating subsequent structural uncertainty analysis. The parallelepiped model is applied to structural uncertainty propagation analysis, and the response interval of the structure is obtained in the case of uncertain initial parameters. Finally, the method described in this paper was applied to several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Hourglass stabilization and the virtual element method

Abstract: A.C. gratefully acknowledges support from the College of Science and Engineering of the University of Leicester and the support of the EPSRC (grant EP/L022745/1). G.M. gratefully acknowledges the support of the LDRD-ER project #20140270 ‘From the finite element method to the virtual element method’ at the Los Alamos National Laboratory and of the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research. A.R. gratefully acknowledges the research support from the University of Milano–Bicocca. N.S. gratefully acknowledges the research support of the National Science Foundation through contract grant CMMI-1334783 to the University of California at Davis.

An enhanced Craig–Bampton method

Abstract: SUMMARY In this paper, we propose a new component mode synthesis method by enhancing the Craig–Bampton (CB) method. To develop the enhanced CB method, the transformation matrix of the CB method is enhanced considering the effect of residual substructural modes and the unknown eigenvalue in the enhanced transformation matrix is approximated by using O’Callahan’s approach in Guyan reduction. Using the newly defined transformation matrix, original finite element models can be more accurately approximated by reduced models. For this reason, the accuracy of the reduced models is significantly improved with a low additional computational cost. We here present the formulation details of the enhanced CB method and demonstrate its performance through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Polygonal finite elements for finite elasticity

Abstract: SUMMARY Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non-convex storedenergy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement-based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly-complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically-based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations. Copyright © 2014 John Wiley & Sons, Ltd.

The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis

Abstract: In this contribution a mortar-type method for the coupling of non-conforming NURBS surface patches is proposed. The connection of non-conforming patches with shared degrees of freedom requires mutual refinement, which propagates throughout the whole patch due to the tensor-product structure of NURBS surfaces. Thus, methods to handle non-conforming meshes are essential in NURBS-based isogeometric analysis. The main objective of this work is to provide a simple and efficient way to couple the individual patches of complex geometrical models without altering the variational formulation. The deformations of the interface control points of adjacent patches are interrelated with a master-slave relation. This relation is established numerically using the weak form of the equality of mutual deformations along the interface. With the help of this relation the interface degrees of freedom of the slave patch can be condensated out of the system. A natural connection of the patches is attained without additional terms in the weak form. The proposed method is also applicable for nonlinear computations without further measures. Linear and geometrical nonlinear examples show the high accuracy and robustness of the new method. A comparison to reference results and to computations with the Lagrange multiplier method is given.

Adaptive h‐refinement for reduced‐order models

Abstract: Our work presents a method to adaptively refine reduced-order models a posteriori without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive h-refinement: it enriches the reduced-basis space online by ‘splitting’ a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive k-means clustering of the state variables using snapshot data. This method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Furthermore, it enables the reduced-order model to satisfy any prescribed error tolerance regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.

A stabilized finite element formulation for monolithic thermo‐hydro‐mechanical simulations at finite strain

Abstract: Summary An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo-hydro-mechanical behavior of porous media undergoing large deformation. We first formulate a finite-deformation thermo-hydro-mechanics field theory for non-isothermal porous media. Projection-based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two-fold inf-sup condition of the equal-order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non-isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo-hydro-mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.

An unbiased computational contact formulation for 3D friction

Abstract: to high accuracy, as is shown. A new 3D friction formulation is also proposed that is a direct extension of the 1D setup, expressing the friction variables in the parameter space used for the curvilinear surface description. The new formulation resorts to classical expressions in the continuum limit. The current approach uses C 1 -smooth contact surface representations based on either Hermite or NURBS interpolation. A penalty regularization is considered for the impenetrability and tangential sticking constraints. The new, unbiased friction formulation is illustrated by several 2D and 3D examples, which include an extensive analysis of the model parameters, a convergence study and the comparison with a classical biased master/slave contact algorithm.

Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes

Abstract: We present a robust method for generating high-order nodal tetrahedral curved meshes. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high-order mesh. The smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh. This means that whenever possible, the resulting mesh preserves the geometrical features of the initial linear mesh such as shape, stretching and size. We present several examples to illustrate the performance of the proposed algorithm. Furthermore, the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. The meshes obtained are suitable for high-order finite element analyses.

Time-domain hybrid formulations for wave simulations in three-dimensional PML-truncated heterogeneous media

Abstract: SUMMARY We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly-matched-layer-(PML)-truncated media. We extend in three dimensions a recently developed twodimensional formulation, by treating the PML via an unsplit-field, but mixed-field, displacement-stress formulation, which is then coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient hybrid scheme. The hybrid treatment leads to, at most, third-order in time semi-discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi-axial PML. We discuss several time-marching schemes, which can be used a la carte, depending on the application: (a) an extended Newmark scheme for third-order in time, either unsymmetric or fully symmetric semi-discrete forms; (b) a standard implicit Newmark for the second-order, unsymmetric semi-discrete forms; and (c) an explicit Runge–Kutta scheme for a first-order in time unsymmetric system. The latter is well-suited for large-scale problems on parallel architectures, while the second-order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.

A face-oriented stabilized Nitsche-type extended variational multiscale method for incompressible two-phase flow

Abstract: A novel extended variational multiscale method for incompressible two-phase flow is proposed. In this approach, the level-set method, which allows for accurately representing complex interface evolutions, is combined with an extended finite element method for the fluid field. Sharp representation of the discontinuities at the interface related to surface-tension effects and large material-parameter ratios are enabled by this approach. To capture the discontinuities, jump enrichments are applied for both velocity and pressure field. Nitsche’s method is then used to weakly impose the continuity of the velocity field. For a stable formulation on the entire domain, residual-based variational multiscale terms are supported by appropriate face-oriented ghost-penalty and fluid stabilization terms in the region of enriched elements. Both face-oriented terms as well as interfacial terms related to Nitsche’s method are introduced such that it is accounted for viscous- and convection-dominated transient flows. As a result, stability and wellconditioned systems are guaranteed independent of the interface location. The proposed method is applied to four numerical examples of increasing complexity: two-dimensional Rayleigh-Taylor instabilities, a twodimensional collapsing water column, three-dimensional rising bubbles as well as a three-dimensional bubble coalescence. Excellent agreement with either analytical solutions or numerical and experimental reference data as well as robustness for all flow regimes is demonstrated for all examples.

A robust Nitsche's formulation for interface problems with spline‐based finite elements

Abstract: Summary The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving embedded interface problems. With a few exceptions, the X-FEM has mostly been used in conjunction with piecewise-linear shape functions and an associated piecewise-linear geometrical representation of interfaces. In the current work, the use of spline-based finite elements is examined along with a Nitsche technique for enforcing constraints on an embedded interface. To obtain optimal rates of convergence, we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces. We further propose a novel weighting for the interfacial consistency terms arising in the Nitsche variational form with B-splines. A qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions. We demonstrate the accuracy and robustness of the proposed method through several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Surrogate modeling of multiscale models using kernel methods

Abstract: This work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods The key element is to learn the interface between the different scales using a fast surrogate for the microscale model, which is given by multivariate kernel expansions The expansions are computed using statistically representative samples of in- and output of the microscale model As learning methods we apply both support vector machines and a vectorial kernel greedy algorithm We demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines First, we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model, and second, a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

Equivalent polynomials for quadrature in Heaviside function enriched elements

Abstract: Summary One of the advantages of partition-of-unity FEMs, like the extended FEM, is the ability of modeling discontinuities independent of the mesh structure. The enrichment of the element functional space with discontinuous or non-differentiable functions requires, when the element stiffness is computed, partitioning into subdomains for quadrature. However, the arbitrary intersection between the base mesh and the discontinuity plane generates quadrature subdomains of complex shape. This is particularly true in three-dimensional problems, where quite sophisticate methodologies have been presented in the literature for the element stiffness evaluation. The present work addresses the problem of Heaviside function enrichments and is based on the replacement of the discontinuous enrichment function with the limit of an equivalent polynomial defined on the entire element domain. This allows for the use of standard Gaussian quadrature in the elements crossed by the discontinuity. The work redefines conceptually the first version of the equivalent polynomial methodology introduced in 2006, allowing a much broader applicability. As a consequence, equivalent polynomials can be computed for all continuum element families in one, two, and three dimensions. Copyright © 2014 John Wiley & Sons, Ltd.

An isogeometric locking-free NURBS-based solid-shell element for geometrically nonlinear analysis

Abstract: With the introduction of IsoGeometric Analysis (IGA) [1], the calculation of shell-type structures has become possible using the exact geometry of the structure regardless of the mesh density. For that, Lagrange polynomials are replaced by NURBS functions, the most commonly used technology in Computer-Aided Design, to perform the analysis. In addition, NURBS functions have a higher order of continuity at knots, which leads to higher per-degree-of-freedom accuracy of the shell solution than with classical Finite Elements Analysis (FEA). However, NURBS elements are likely to suffer from the same locking problems as classical elements. Then, to really benefit from NURBS, some strategies need to be implemented to answer the locking issue.

Analysis of an HDG method for linear elasticity

Abstract: Summary We present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058–1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewise-polynomial approximations of degree k≥0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k + 1/2, and the displacement converges with order k + 1. We also provide numerical results showing that the orders of convergence are actually sharp. Copyright © 2014 John Wiley & Sons, Ltd.

Constrained pseudo‐transient continuation

Abstract: This paper addresses the problem of finding a stationary point of a nonlinear dynamical system whose state variables are under inequality constraints. Systems of this type often arise from the discretization of partial differential equations that model physical phenomena (e.g. fluid dynamics) in which the state variables are under realizability constraints (e.g. positive pressure and density). We start from the popular pseudotransient continuation method and augment it with nonlinear inequality constraints. The constraint handling technique does not help in situations where no steady-state solution exists, for example due to an underresolved discretization of partial differential equations. However, an often overlooked situation is one in which the steady-state solution exists but cannot be reached by the solver, which typically fails due to the violation of constraints, i.e. a non-physical state error during state iterations. This is the shortcoming that we address by incorporating physical realizability constraints into the solution path from the initial condition to steady state. While we focus on the discontinuous Galerkin method applied to fluid dynamics, our technique relies only on implicit time marching and hence can be extended to other spatial discretizations and other physics problems. We analyze the sensitivity of the method to a range of input parameters and present results for compressible turbulent flows that show that the constrained method is significantly more robust than a standard unconstrained method while on par in terms of cost. Copyright © 0000 John Wiley & Sons, Ltd.

A monolithic geometric multigrid solver for fluid‐structure interactions in ALE formulation

Abstract: Summary We present a monolithic geometric multigrid solver for fluid-structure interaction problems in Arbitrary Lagrangian Eulerian coordinates. The coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill-conditioned systems of algebraic equations. Direct solvers usually are out of question because of memory limitations, and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices. The use of partitioned preconditioners in Krylov subspace iterations is an option, but the convergence will be limited by the outer partitioning. Our proposed solver is based on a Newton linearization of the fully monolithic system of equations, discretized by a Galerkin finite element method. Approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration, preconditioned by a geometric multigrid solver. The special character of fluid-structure interactions is accounted for by a partitioned scheme within the multigrid smoother only. Here, fluid and solid field are segregated as Dirichlet–Neumann coupling. We demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems. While 2d problems are well manageable with available direct solvers, challenging 3d problems highly benefit from the resulting multigrid solver. Copyright © 2015 John Wiley & Sons, Ltd.