Adaptive component mode synthesis in linear elasticity

Jakobsson, Håkan; Bengzon, Fredrik; Larson, Mats G.

doi:10.1002/nme.3078

Cited by 25 publications

(34 citation statements)

References 19 publications

(22 reference statements)

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“…Within the CMS approach this is realized by utilizing an eigenmode expansion [6,18,19,26], which has recently been combined with input-output-based model reduction in [20]. In [11] Eftang and Patera develop an empirical pairwise training procedure for port reduction within the scRBE context: Modes are selected from traces of snapshots generated by random boundary conditions.…”

mentioning

confidence: 99%

Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

Smetana¹,

Patera²

2016

SIAM J. Sci. Comput.

111

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Abstract. In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or "ports," in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter-dependent setting and to construct a quasi-optimal parameter-independent port space. Moreover, it is shown that, given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields an sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity, although the idea may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and of the sc approximation based on spectral modes for nonseparable and irregular geometries such as an I-beam with an internal crack. 1. Introduction. In the last decades numerical simulations based on partial differential equations (PDEs) have significantly gained importance in engineering applications. However, both the geometric complexity of the considered structures, such as ships, aircraft, and turbines, and the intricacy of the simulated physical phenomena often make a straightforward application of, say, the finite element (FE) method prohibitive. This is particularly true if multiple simulation requests or a real-time simulation response is desired, as in engineering design and optimization.One way to tackle such complex problems is to exploit the natural decomposition of the structures into components and apply static condensation (sc) to obtain a (Schur complement) system of the size of the degrees of freedom (DOFs) on all interfaces or ports in the system. To mitigate the computational costs for the required PDE solvers in the interior of the component, model order reduction procedures may be applied. One popular approach is component mode synthesis (CMS), introduced in [4,21], which uses an approximation based on the eigenmodes of local constrained eigenvalue problems. The static condensation reduced basis element (scRBE) method [22,23]

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confidence: 99%

Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

Smetana¹,

Patera²

2016

SIAM J. Sci. Comput.

111

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“…We would not in general expect a similar result for the "classical" non-singular Sturm-Liouville choice s m,j = 1, and we note that port reduction approaches within the CMS framework [6,18] typically consider regular rather than singular eigenproblems. Also note that in the case y m,j = 0 a solution to (36) is always (κ 1 m,j = 0, τ 1 m,j = constant), and hence χ m,j,1 is constant for any y m,j (recall in the case y m,j > 0 we set χ m,j,1 = ρ m,j,1 , which is chosen constant).…”

Section: Port Approximationmentioning

confidence: 78%

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

104

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We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces -the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system, and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditionerthe error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.

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“…In order to obtain a priori estimates for the error on the displacements, error estimation methods have been developed (see e.g. [28][29][30]). …”

Section: Expansion Of Superelement Resultsmentioning

confidence: 99%

Dynamic Models for Load Calculation Procedures of Offshore Wind Turbine Support Structures: Overview, Assessment, and Outlook

Valk

Voormeeren

Valk

et al. 2015

Journal of Computational and Nonlinear Dynamics

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One of the main mechanisms for driving down the cost of offshore wind energy is to install ever larger wind turbines in larger wind farms. At the same time, these turbines are placed further offshore in deeper waters. As a result traditional monopile foundations are not always feasible and multi-membered foundations, such as jackets and tripods, are required.Typically thousands of load cases need to be simulated for the design and certification of offshore wind turbines. As models of such foundations are significantly larger than their monopile counterparts, model reduction is often applied to limit the computational costs. Additionally, the foundation design is generally done by a specialized company, which bases its design on the results of the load simulations. Hence, an accurate estimation of the stresses in load simulation is essential to predict the integrity and the lifetime of different designs. The effect on the load accuracy of both the model reduction as well as the post-processing method used by foundation designers are investigated in this paper. A case study is performed on a jacket-based wind turbine model to verify and quantify the findings.Firstly, it is observed that the effect of the reduced foundation model on the wind turbine loads is negligible. However, both the reduction method and the post-processing method applied by the foundation designer have a large influence on the fatigue loading in the jacket. It is shown that the popular Guyan reduction results in significant errors on the fatigue damage and that a static post-processing analysis leads to serious underestimations of the fatigue loads. Finally, an outlook is given into future developments in the field of load calculations for offshore wind turbine foundation design. Nomenclature B Signed Boolean operator C Damping matrix D el i Fatigue damage in element i f Array of external forces g Array of connection forces Journal of Computational and Nonlinear Dynamics. Downloaded From: http://computationalnonlinear.asmedigitalcollection.asme.org/ on 02/20/2015 Terms of Use: http://asme.org/terms A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dH s Significant wave height K Stiffness matrix L Boolean assembly operator M Mass matrix p Array of nonlinear internal damping and elastic forces q Array of reduced degrees of freedom R Reduction matrix r Array of residual forces T Orthogonal projector u Array of displacement degrees of freedom v a Average wind speeds α Array of modal force amplitudes η Array of modal amplitudes λ Array of Lagrange multipliers φ Normal mode ψ C Constraint mode ω Frequency [i] Associated to internal DoF [b] Associated to boundary DoF [W T ] Associated to the wind turbine substructure [F] Associated to the foundation substructurė First time derivativë Second time derivativē Primal assembled Reduced variable CB Craig-Bampton method CMS Component mode synthesis DC Displacement controlled DEL Damage equivalent load DLC Design load case DoF Degree of freedom FC Force controlled FD Foundation designer GR Guyan red...

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Adaptive component mode synthesis in linear elasticity

Cited by 25 publications

References 19 publications

Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Dynamic Models for Load Calculation Procedures of Offshore Wind Turbine Support Structures: Overview, Assessment, and Outlook

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