Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Abbasi, Mohammad Hossein; Iapichino, Laura; Besselink, Bart; Schilders, W.H.A.; Wouw, Nathan van de

doi:10.1016/j.cma.2019.112688

Cited by 6 publications

(11 citation statements)

References 37 publications

(125 reference statements)

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“…To handle the nonlinearities at the boundaries, the following RB ansatz is used by lifting the reduced solution 7

\begin{align} {bold-italicÛ}^{n} false(μ false) & = {bold-italicU}_{B}^{n} + true \sum_{i = 1}^{N} a_{i}^{n} false(μ false) ϕ_{i}, \\ ϕ_{i} {false|}_{x = false{0, L false}} & = 0, \end{align}

where

U_{B}^{n}

is a vector that enables the RB solution

Û^{n} (μ)

to satisfy the boundary conditions (

Û

can be either

\hat{ρ}

\hat{m}

). In other words, the vector

U_{B}^{n}

encodes the exact satisfaction of the boundary conditions.…”

Section: Reduced Order Model Formulationmentioning

confidence: 99%

“…The worst approximation is determined by evaluating the output error between the RB solution of system (22) and the full order solution of (16). An error bound or an error estimate can be useful to drastically alleviate the computational effort of the error exploration among the domain

𝒟_{h}

7 (if the error estimate can be computed at low computational cost, ie, the computation of the error estimate does not scale with the dimension of the full‐order model). Other empirical error estimates can also be useful for nonlinear problems, such as those in References 18, 19 .…”

Section: Reduced Order Model Formulationmentioning

confidence: 99%

“…Operators such as

C C_{0} \in R^{N_{y} \times 2}

are of low dimensions and can be defined in the offline stage. For more information, refer to Reference 7 . The initial conditions for the system are also obtained by a Galekin projection of the initial condition; for instance,

a_{ρ}^{0} = Φ_{ρ}^{T} ρ^{0}

.…”

Section: Reduced Order Model Formulationmentioning

confidence: 99%

“…Handling boundary conditions becomes challenging due to the global nature of the RB functions and their independency over time 6 . To resolve this issue, a lifting method 7 is implemented here and tested for an industry‐relevant MPD application. Within this lifting method, the effect of the boundary conditions is segregated from the internal domain and is incorporated later as a state dependent and locally acting input.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, to the best of our knowledge, for the first time the RB method is applied to a drilling system with MPD and the ansatz introduced in Reference 7 is tested for dealing with implicit and nonlinear boundary conditions governing the hydraulics dynamics of the drilling system. Moreover, a new methodology is introduced to automatically capture the local nonlinearities inside the computational domain by enriching RB functions with specific local basis functions.…”

Section: Introductionmentioning

confidence: 99%

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Reduced basis method for managed pressure drilling based on a model with local nonlinearities

Abbasi

Iapichino

Lordejani

et al. 2020

Numerical Meth Engineering

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To circumvent restrictions of conventional drilling methods, such as slow control actions and inability to drill depleted reservoirs, a drilling method called managed pressure drilling (MPD) has been developed. In MPD, single‐phase flow processes can be modeled as a feedback interconnection of a high‐order linear system and a low‐order nonlinear system. These nonlinearities appear locally both inside and at the boundaries of the computational domain. To obtain a fast simulation platform for real‐time purposes (eg, online model‐based controller implementation), model order reduction is required for MPD. However, the local nonlinearities render applying model order reduction techniques challenging. In this study, a new approach is proposed to deal with such nonlinearities within the reduced basis (RB) context and it is successfully tested on a model for MPD. Contrary to the classical RB technique, the proposed approach not only does not generate nonphysical spikes at the locations of these local nonlinearities but also yields high speedup factors. The obtained reduced‐order model can be used for efficient online simulation and controller design for drilling systems with MPD.

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“…To handle the nonlinearities at the boundaries, the following RB ansatz is used by lifting the reduced solution 7

\begin{align} {bold-italicÛ}^{n} false(μ false) & = {bold-italicU}_{B}^{n} + true \sum_{i = 1}^{N} a_{i}^{n} false(μ false) ϕ_{i}, \\ ϕ_{i} {false|}_{x = false{0, L false}} & = 0, \end{align}

where

U_{B}^{n}

is a vector that enables the RB solution

Û^{n} (μ)

to satisfy the boundary conditions (

Û

can be either

\hat{ρ}

\hat{m}

). In other words, the vector

U_{B}^{n}

encodes the exact satisfaction of the boundary conditions.…”

Section: Reduced Order Model Formulationmentioning

confidence: 99%

𝒟_{h}

Section: Reduced Order Model Formulationmentioning

confidence: 99%

“…Operators such as

C C_{0} \in R^{N_{y} \times 2}

a_{ρ}^{0} = Φ_{ρ}^{T} ρ^{0}

.…”

Section: Reduced Order Model Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Reduced basis method for managed pressure drilling based on a model with local nonlinearities

Abbasi

Iapichino

Lordejani

et al. 2020

Numerical Meth Engineering

Self Cite

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show abstract

Well‐scaled, a‐posteriori error estimation for model order reduction of large second‐order mechanical systems

2020

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Model Order Reduction is used to vastly speed up simulations but it also introduces an error to the simulation results, which needs to be controlled. The a‐posteriori error estimator of Ruiner et al. for second‐order systems, which is based on the residual, has the advantage of having provable upper bounds and being usable independently of the reduction method. Nevertheless a bottleneck is found in the offline phase, making it unusable for larger models. We use the spectral theorem, power series expansions, monotonicity properties, and self‐tailored algorithms to largely speed up the offline phase by one polynomial order both in terms of computation time as well as storage complexity. All properties are proven rigorously. This eliminates the aforementioned bottleneck. Hence, the error estimator of Ruiner et al. can finally be used for large, linear, second‐order mechanical systems reduced by any model reduction method based on Petrov–Galerkin reduction. The examples show speedups of up to 28.000 and the ability to compute much larger systems with a fixed amount of memory.

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A data-based stability-preserving model order reduction method for hyperbolic partial differential equations

et al. 2022

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Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions

Abstract: van de Wouw, N. (2020). Error estimation in reduced basis method for systems with time-varying and nonlinear boundary conditions. Computer Methods in Applied Mechanics and Engineering, 360, [112688].

Cited by 6 publications

References 37 publications

Reduced basis method for managed pressure drilling based on a model with local nonlinearities

Reduced basis method for managed pressure drilling based on a model with local nonlinearities

Well‐scaled, a‐posteriori error estimation for model order reduction of large second‐order mechanical systems

A data-based stability-preserving model order reduction method for hyperbolic partial differential equations

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