Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems

In this paper we develop adaptive iterative coupling schemes for the Biot system modeling coupled poromechanics problems. We particularly consider the space-time formulation of the fixed-stress iterative scheme, in which we first solve the problem of flow over the whole space-time interval, then exploiting the space-time information for solving the mechanics. Two common discretizations of this algorithm are then introduced based on two coupled mixed finite element methods in-space and the backward Euler scheme in-time. Therefrom, adaptive fixed-stress algorithms are build on conforming reconstructions of the pressure and displacement together with equilibrated flux and stresses reconstructions. These ingredients are used to derive a posteriori error estimates for the fixed-stress algorithms, distinguishing the different error components, namely the spatial discretization, the temporal discretization, and the fixed-stress iteration components. Precisely, at the iteration $k\geq 1$ of the adaptive algorithm, we prove that our estimate gives a guaranteed and fully computable upper bound on the energy-type error measuring the difference between the exact and approximate pressure and displacement. These error components are efficiently used to design adaptive asynchronous time-stepping and adaptive stopping criteria for the fixed-stress algorithms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive iterative coupling algorithms

A posteriori error estimation and modeling of unsaturated flow in fractured porous media

This doctoral thesis focuses on three topics: (1) modeling of unsaturated flow in fractured porous media, (2) a posteriori error estimation for mixed-dimensional elliptic equations, and (3) contributions to open-source software for complex multiphysics processes in porous media. In our first contribution, following a Discrete-Fracture Matrix (DFM) approach, we propose a model where Richards' equation governs the water flow in the matrix, whereas fractures are represented as lower-dimensional open channels, naturally providing a capillary barrier to the water flow. Therefore, water in the matrix is only allowed to imbibe the fracture if the capillary barrier is overcome. When this occurs, we assume that the water inside the fracture flows downwards without resistance and, therefore, is instantaneously at hydrostatic equilibrium. This assumption can be justifiable for fractures with sufficiently large apertures, where capillary forces play no role. Mathematically, our model can be classified as a coupled PDE-ODE system of equations with variational inequalities, in which each fracture is considered a potential seepage face. Our second contribution deals with error estimation for mixed-dimensional (mD) elliptic equations, which, in particular, model single-phase flow in fractured porous media. Here, based on the theory of functional a posteriori error estimates, we derive guaranteed upper bounds for the mD primal and mD dual variables, and two-sided bounds for the mD primal-dual pair. Moreover, we improve the standard results of the functional approach by proposing four ways of estimating the residual errors based on the conservation properties of the approximations, that is, (1) no conservation, (2) subdomain conservation, (3) local conservation, and (4) pointwise conservation. This results in sharper and fully-computable bounds when mass is conserved either locally or exactly. To our knowledge, to date, no error estimates have been available for fracture networks, including fracture intersections and floating subdomains. Our last contribution is related to the development of open-source software. First, we present the implementation of a new multipoint finite-volume-based module for unsaturated poroelasticity, compatible with the Matlab Reservoir Simulation Toolbox (MRST). Second, we present a new Python-based simulation framework for multiphysics processes in fractured porous media, named PorePy. PorePy, by design, is particularly well-suited for handling mixed-dimensional geometries, and thus optimal for DFM models. The first two contributions discussed above were implemented in PorePy.Denne avhandlingen tar for seg tre emner: (1) modellering av flyt i umettet porøst medium med sprekker, (2) a posteriori feilestimater for blandet-dimensjonale elliptiske ligninger, og (3) bidrag til åpen kildekode for komplekse multifysikk-prosesser i porøse medier. I det første bidraget anvender vi en Discrete-Fracture Matrix (DFM) (Diskret-Sprekk Matrise) metode til å sette opp en modell hvor Richard's ligning modellerer vann-flyt i matrisen, og sprekkene representeres som lavere-dimensjonale åpne kanaler, som naturlig virker som kapillærbarrierer til vann-flyten. Derfor vil vann i matrisen kun få tilgang til sprekken når kapillærbarrieren blir brutt. Når det inntreffer, antar vi at vannet i sprekken flyter nedover uten motstand, og at hydrostatisk ekvilibrium derfor inntreffer øyeblikkelig. Slike antakelser kan rettferdiggjøres for sprekker med tilstrekkelig stor apertur (åpning), hvor kapillærkrefter ikke har noen innvirkning. Fra et matematisk standpunkt kan modellen klassifiseres som en sammenkoblet PDE-ODE med variasjonelle ulikheter hvor hver sprekk behandles som en filtreringsfase. Det andre bidraget tar for seg feilestimater for blandet-dimensjonale elliptiske ligninger, som modellerer en-fase flyt i porøse medier med sprekker. Her anvender vi teorien for "funksjonal a posteriori feilestimater" til å finne øvre skranker for primær og dual variablene, samt øvre og nedre skranker for primær-dual paret. Dessuten viser vi at vi kan forbedre standardresultatene fra "funksjonal a posteriori feilestimater" ved å foreslå fire måte å estimere residualfeilen basert på bevaringsegenskapene til diskretiseringen. De fire forskjellige bevaringsegenskapene er; ingen bevaringsegenskap, under- domene bevaring, lokal bevaring og punktvis bevaring. Dette fører til skarpere skranker som er mulige å beregne når masse er bevart enten lokalt, eller eksakt. Vi kjenner ikke til andre tilgjengelige feilestimater for sprekknettverk som inkluderer snitt av sprekker og sprekkrender som ligger innenfor domenets rand. Det siste bidraget omhandler utvikling av åpen kildekode. Først presenterer vi imple- menteringen av en multipunktfluks-basert modul for flyt i umettet deformerbart porøst medium som er kompatibelt med "Matlab Reservoir Simulation Toolbox"(MRST). I tillegg presenterer vi et nytt Python-basert rammeverk for simulering av multifysikkprosesser i porøse medier med sprekker, som heter PorePy. Dette rammeverket er designet for å håndtere geometrier med blandede dimensjoner og er derfor optimalt for DFM modeller. De to første bidragene i avhandlingen (nevnt over) er implementert i PorePy.Doktorgradsavhandlin

Discontinuous Finite Element Methods for Interface Problems: Robust A Priori and A Posteriori Error Estimates

For elliptic interface problems in two and three dimensions, this paper studies a priori and residual-based a posteriori error estimations for the Crouzeix–Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with no assumption on the distribution of the diffusion coefficient

Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media

A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table

Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery

[EN] Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.This work was supported by the EPSRC Grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has received partial support from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. All authors also thank the partial support of the Framework Programme 7 Initial Training Network Funding under Grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology."González Estrada, OA.; Nadal Soriano, E.; Ródenas, J.; Kerfriden, P.; Bordas, S.; Fuenmayor Fernández, FJ. (2014). Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Computational Mechanics. 53(5):957-976. https://doi.org/10.1007/s00466-013-0942-8S957976535Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Wiley, ChichesterBabuška I, Rheinboldt WC (1978) A-posteriori error estimates for the finite element method. Int J Numer Methods Eng 12(10):1597–1615Cottereau R, Díez P, Huerta A (2009) Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method. Comput Mech 44(4):533–547Larsson F, Hansbo P, Runesson K (2002) Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int J Numer Methods Eng 55(8):879–894Díez P, Parés N, Huerta A (2003) Recovering lower bounds of the error by postprocessing implicit residual a posteriori error estimates. Int J Numer Methods Eng 56(10):1465–1488Verfürth R (1996) A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley, ChichesterStein E, Ramm E, Rannacher R (2003) Error-controlled adaptive finite elements in solid mechanics. Wiley, ChichesterDíez P, Egozcue JJ, Huerta A (1998) A posteriori error estimation for standard finite element analysis. Comput Methods Appl Mech Eng 163(1–4):141–157Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24(2):337–357Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int J Numer Methods Eng 33(7):1331–1364Bordas SPA, Duflot M (2007) Derivative recovery and a posteriori error estimate for extended finite elements. Comput Methods Appl Mech Eng 196(35–36):3381–3399Bordas SPA, Duflot M, Le P (2008) A simple error estimator for extended finite elements. Commun Numer Methods Eng 24(11):961–971Ródenas JJ, González-Estrada OA, Tarancón JE, Fuenmayor FJ (2008) A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting. Int J Numer Methods Eng 76(4):545–571Duflot M, Bordas SPA (2008) A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Methods Eng 76:1123–1138Prange C, Loehnert S, Wriggers P (2012) Error estimation for crack simulations using the XFEM. Int J Numer Methods Eng 91:1459–1474Hild P, Lleras V, Renard Y (2010) A residual error estimator for the XFEM approximation of the elasticity problem. Comput Mech (submitted)González-Estrada O, Natarajan S, Ródenas J, Nguyen-Xuan H, Bordas S (2012) Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity. Comput Mech 52(1):37–52González-Estrada O, Ródenas J, Bordas S, Duflot M, Kerfriden P, Giner E (2012) On the role of enrichment and statical admissibility of recovered fields in a-posteriori error estimation for enriched finite element methods. Eng Comput 29(8):2–2Díez P, Ródenas JJ, Zienkiewicz OC (2007) Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error. Int J Numer Methods Eng 69(10):2075–2098Ródenas JJ, González-Estrada OA, Díez P, Fuenmayor FJ (2010) Accurate recovery-based upper error bounds for the extended finite element framework. Comput Methods Appl Mech Eng 199(37–40):2607–2621Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl 41(5–6):735–756Ladevèze P (2006) Upper error bounds on calculated outputs of interest for linear and nonlinear structural problems. Comptes Rendus Mécanique 334(7):399–407Ladevèze P, Pelle JP (2005) Mastering calculations in linear and nonlinear mechanics. Mechanical engineering series. Springer, BerlinRüter M, Stein E (2006) Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput Methods Appl Mech Eng 195(4–6):251–278Ladevèze P, Rougeot P, Blanchard P, Moreau JP (1999) Local error estimators for finite element linear analysis. Comput Methods Appl Mech Eng 176(1–4):231–246de Almeida JPM, Pereira OJBA (2006) Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems. Comput Methods Appl Mech Eng 195(4–6):279–296Cirak F, Ramm E (1998) A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Comput Methods Appl Mech Eng 156(1–4):351–362Verfürth R (1999) A review of a posteriori error estimation techniques for elasticity problems. Comput Methods Appl Mech Eng 176:419–440Zienkiewicz OC, Taylor R (2000) The finite element method: the basis, vol 1, 5th edn. Butterworth-Heinemann, OxfordLadevèze P, Leguillon D (1983) Error estimate procedure in the finite element method and applications. SIAM J Numer Anal 20(3):485–509Pled F, Chamoin L, Ladevèze P (2012) An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses. Comput Mech 49(3):357–378Blacker T, Belytschko T (1994) Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements. Int J Numer Methods Eng 37(3):517–536González-Estrada OA, Nadal E, Ródenas JJ, Kerfriden P, Bordas SPA (2012) Error estimation in quantities of interest for XFEM using recovery techniques. In: Yang ZJ (ed)20th UK National Conference of the Association for Computational Mechanics in Engineering (ACME). The University of Manchester, Manchester, pp 333–336Pannachet T, Sluys LJ, Askes H (2009) Error estimation and adaptivity for discontinuous failure. Int J Numer Methods Eng 78(5):528–563Ródenas JJ, Tur M, Fuenmayor FJ, Vercher A (2007) Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique. Int J Numer Methods Eng 70(6):705–727Szabó BA, Babuška I (1991) Finite element analysis. Wiley, New YorkGiner E, Tur M, Fuenmayor FJ (2009) A domain integral for the calculation of generalized stress intensity factors in sliding complete contacts. Int J Solids Struct 46(3–4):938–951Ródenas JJ (2005) Goal Oriented Adaptivity: Una introducción a través del problema elástico lineal. Technical Report, CIMNE, PI274, Barcelona, SpainGonzález-Estrada OA, Ródenas JJ, Nadal E, Bordas SPA, Kerfriden P (2011) Equilibrated patch recovery for accurate evaluation of upper error bounds in quantities of interest. Adaptive Modeling and Simulation. In: Aubry D, Díez P, Tie B, Parés N (eds) Proceedings of V ADMOS 2011. CIMNE, ParisVerdugo F, Díez P, Casadei F (2011) Natural quantities of interest in linear elastodynamics for goal oriented error estimation and adaptivity. Adaptive Modeling and Simulation. In: Aubry D, Díez P, Tie B, Parés N (eds) Proceedings of V ADMOS 2011. CIMNE, ParisRódenas JJ, Giner E, Tarancón JE, González-Estrada OA (2006) A recovery error estimator for singular problems using singular+smooth field splitting. In: Topping BHV, Montero G, Montenegro R (eds) Fifth International Conference on Engineering Computational Technology. Civil-Comp Press, Stirling, ScotlandNadal E, Ródenas JJ, Tarancón JE, Fuenmayor FJ (2011) On the advantages of the use of cartesan grid solvers in shape optimization problems. Adaptive Modeling and Simulation. In: Aubry D, Díez P, Tie B, Parés N (eds) Proceedings of V ADMOS 2011. CIMNE, ParisNadal E, Ródenas J, Albelda J, Tur M, Fuenmayor FJ (2013) Efficient Finite Element methodology based on Cartesian Grids. Application to structural shape optimization. Abstr Appl Anal 2013(Article ID 953786):1–19.Moumnassi M, Belouettar S, Béchet E, Bordas SPA, Quoirin D, Potier-Ferry M (2011) Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces. Comput Methods Appl Mech Eng 200(5–8):774–796Belytschko T, Parimi C, Moës N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56(4):609–635Moës N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192(28–30):3163–3177Gordon WJ, Hall CA (1973) Construction of curvilinear co-ordinate systems and applications to mesh generation. Int J Numer Methods Eng 7(4):461–477Sevilla R, Fernández-Méndez S (2008) NURBS-enhanced finite element method ( NEFEM ). Online 76(1):56–83Béchet E, Moës N, Wohlmuth B (2009) A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method. Int J Numer Methods Eng 78:931–954Schweitzer M (2011) Stable enrichment and local preconditioning in the particle-partition of unity method. Numerische Mathematik 118(1):137–170Menk A, Bordas S (2011) A robust preconditioning technique for the extended finite element method. Int J Numer Methods Eng 85(13):1609–1632Hiriyur B, Tuminaro R, Waisman H, Boman E, Keyes D (2012) A quasi-algebraic multigrid approach to fracture problems based on extended finite elements. SIAM J Sci Comput 34(2):603–626Gerstenberger A, Tuminaro R (2012) An algebraic multigrid approach to solve xfem based fracture problems. Int J Numer Methods Eng 94(3):248–272Ladevèze P, Marin P, Pelle JP, Gastine JL (1992) Accuracy and optimal meshes in finite element computation for nearly incompressible materials. Comput Methods Appl Mech Eng 94(3):303–315Coorevits P, Ladevèze P, Pelle JP (1995) An automatic procedure with a control of accuracy for finite element analysis in 2D elasticity. Comput Methods Appl Mech Eng 121:91–120Li LY, Bettess P (1995) Notes on mesh optimal criteria in adaptive finite element computations. Commun Numer Methods Eng 11(11):911–915Fuenmayor FJ, Oliver JL (1996) Criteria to achieve nearly optimal meshes in the h-adaptive finite element method. Int J Numer Methods Eng 39:4039–4061Abel JF, Shephard MS (1979) An algorithm for multipoint constraints in finite element analysis. Int J Numer Methods Eng 14(3):464–467Farhat C, Lacour C, Rixen D (1998) Incorporation of linear multipoint constraints in substructure based iterative solvers. Part 1: a numerically scalable algorithm. Int J Numer Methods Eng 43(6):997–1016Bordas SPA, Moran B (2006) Enriched finite elements and level sets for damage tolerance assessment of complex structures. Eng Fract Mech 73(9):1176–1201Bordas S, Conley J, Moran B, Gray J, Nichols E (2007) A simulation-based design paradigm for complex cast components. Eng Comput 23(1):25–37Bordas SPA, Nguyen PV, Dunant C, Guidoum A, Nguyen-Dang H (2007) An extended finite element library. Int J Numer Methods Eng 71(6):703–732Wyart E, Coulon D, Duflot M, Pardoen T, Remacle JF, Lani F (2007) A substructured FE-shell/XFE-3D method for crack analysis in thin-walled structures. Int J Numer Methods Eng 72(7):757–779Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle JF, Lani F (2008) Substructuring FEXFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 215(2):626–63

Enabling Automated, Reliable and Efficient Aerodynamic Shape Optimization With Output-Based Adapted Meshes

Simulation-based aerodynamic shape optimization has been greatly pushed forward during the past several decades, largely due to the developments of computational fluid dynamics (CFD), geometry parameterization methods, mesh deformation techniques, sensitivity computation, and numerical optimization algorithms. Effective integration of these components has made aerodynamic shape optimization a highly automated process, requiring less and less human interference. Mesh generation, on the other hand, has become the main overhead of setting up the optimization problem. Obtaining a good computational mesh is essential in CFD simulations for accurate output predictions, which as a result significantly affects the reliability of optimization results. However, this is in general a nontrivial task, heavily relying on the user’s experience, and it can be worse with the emerging high-fidelity requirements or in the design of novel configurations. On the other hand, mesh quality and the associated numerical errors are typically only studied before and after the optimization, leaving the design search path unveiled to numerical errors. This work tackles these issues by integrating an additional component, output-based mesh adaptation, within traditional aerodynamic shape optimizations. First, we develop a more suitable error estimator for optimization problems by taking into account errors in both the objective and constraint outputs. The localized output errors are then used to drive mesh adaptation to achieve the desired accuracy on both the objective and constraint outputs. With the variable fidelity offered by the adaptive meshes, multi-fidelity optimization frameworks are developed to tightly couple mesh adaptation and shape optimization. The objective functional and its sensitivity are first evaluated on an initial coarse mesh, which is then subsequently adapted as the shape optimization proceeds. The effort to set up the optimization is minimal since the initial mesh can be fairly coarse and easy to generate. Meanwhile, the proposed framework saves computational costs by reducing the mesh size at the early stages of the optimization, when the design is far from optimal, and avoiding exhaustive search on low-fidelity meshes when the outputs are inaccurate. To further improve the computational efficiency, we also introduce new methods to accelerate the error estimation and mesh adaptation using machine learning techniques. Surrogate models are developed to predict the localized output error and optimal mesh anisotropy to guide the adaptation. The proposed machine learning approaches demonstrate good performance in two-dimensional test problems, encouraging more study and developments to incorporate them within aerodynamic optimization techniques. Although CFD has been extensively used in aircraft design and optimization, the design automation, reliability, and efficiency are largely limited by the mesh generation process and the fixed-mesh optimization paradigm. With the emerging high-fidelity requirements and the further developments of unconventional configurations, CFD-based optimization has to be made more accurate and more efficient to achieve higher design reliability and lower computational cost. Furthermore, future aerodynamic optimization needs to avoid unnecessary overhead in mesh generation and optimization setup to further automate the design process. The author expects the methods developed in this work to be the keys to enable more automated, reliable, and efficient aerodynamic shape optimization, making CFD-based optimization a more powerful tool in aircraft design.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163034/1/cgderic_1.pd