Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

Mirhoseini, Marzieh Alireza; Zahr, Matthew J.

doi:10.1016/j.jcp.2022.111739

Cited by 11 publications

(18 citation statements)

References 62 publications

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“…We let

\boldsymbol{X}\in {\mathbb{R}}^{N_{\boldsymbol{x}}}

denote the nodes of the reference mesh, making

𝒢 (X; X) = X

(the identity map). Details on construction of the domain mapping can be found in Reference 33. Throughout the document, (5) will be referred to as the high‐dimensional model (HDM).…”

Section: Governing Equations and Discretizationmentioning

confidence: 99%

“…In our previous work, 33 we introduced a model reduction method, implicit feature tracking , that constructs a nonlinear approximation manifold by composing a low‐dimensional affine space with a space of bijections of the underlying domain. The reduced‐order model with implicit feature tracking (ROM‐IFT) is defined as a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimizes the residual of the unreduced PDE discretization, which is analogous to standard minimum‐residual reduced‐order models, except our method expands the optimization space to include admissible domain mappings.…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 introduces the governing system of parametrized partial differential equations, its transformation to a fixed reference domain, and the corresponding discretization. Section 3 recalls the implicit feature tracking method of Reference 33 and introduces an empirical quadrature hyperreduction procedure to accelerate assembly of nonlinear terms, a Levenberg‐Marquardt solver for the hyperreduced optimization problems, and a greedy training algorithm. Section 4 demonstrates the merits of the proposed hyperreduction framework for two shock‐dominated benchmarks and Section 5 concludes the paper.…”

Section: Introductionmentioning

confidence: 99%

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Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Mirhoseini,

Zahr

2023

Numerical Methods in Fluids

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SummaryThis work introduces an empirical quadrature‐based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection‐dominated problems with limited training. The proposed approach circumvents the slowly decaying ‐width limitation of linear model reduction techniques applied to convection‐dominated problems by using a nonlinear approximation manifold systematically defined by composing a low‐dimensional affine space with bijections of the underlying domain. The reduced‐order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online‐efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh‐independent operations. The proposed reduced‐order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock‐dominated computational fluid dynamics benchmarks.

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“…We let

\boldsymbol{X}\in {\mathbb{R}}^{N_{\boldsymbol{x}}}

denote the nodes of the reference mesh, making

𝒢 (X; X) = X

(the identity map). Details on construction of the domain mapping can be found in Reference 33. Throughout the document, (5) will be referred to as the high‐dimensional model (HDM).…”

Section: Governing Equations and Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Mirhoseini,

Zahr

2023

Numerical Methods in Fluids

Self Cite

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“…However, the PMOR of a wide range of hyperbolic problems with non-linearity and discontinuity still remain a challenge. Therefore, the researchers within the MOR community are intensively seeking methodologies to reduce the Kolmogorov n-width of the solution manifold [45,46]. Boncoraglio et al [47] efficiently solved multidisciplinary design optimization problem, which blends both linear and nonlinear constraints in aerodynamics using projection-based MOR along with an active manifold.…”

Section: Parametric Model Order Reduction Of Numerical Model 61 Funda...mentioning

confidence: 99%

Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction

Mikhaylenko¹,

Rauter²,

Muralidhar³

et al. 2022

Preprint

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Guided ultrasonic waves are suitable for use in the context of structural health monitoring of thin-walled, plate-like structures. Hence, observing the wave propagation in the plates can provide an indication of whether damage has occurred in the structure. In this work, the wave propagation in fiber metal laminate consisting of thin steel foils and layers of carbon fiber-reinforced polymer is studied, focusing on the main propagation characteristics like dispersion diagrams and displacement fields. For this purpose, the dispersion diagrams derived from the analytical framework and numerical simulations are first determined and compared to each other. Next, the displacement fields are computed using the global matrix method for two excitation frequencies. The results derived from the analytical framework is used to validate numerically determined displacement fields based on a 2D and a 3D modeling approach. For both investigations the results of the analytical treatment and the numerical simulation show good agreement. Furthermore, the displacement field reveals the typical and well-known characteristics of the propagation of guided waves in thin-walled structures. Since the use of full 3D models involves a very high computational cost, this work also successfully investigates the possibility for model order reduction to decrease the computational time and costs of the simulation without the loss of accuracy.

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“…• Another group of researchers pursue to break the Kolmogorov barrier by seeking nonlinear manifolds instead of linear subspaces for the low-rank representations of the dynamics. One class of popular approaches leverage transformation of the subspaces/snapshots to construct such manifolds with the goal of recovering low-rank structures with fast Kolmogorov N-width decays, which include the method of freezing [66], shifted POD [67,68,69], transported subspaces [70] or snapshots [71], and implicit feature tracking [72]. Moreover, other researchers seek to directly compute the nonlinear manifolds via either convolutional autoecoders [73,74] or explicitly quadratic approximation [75,76,77].…”

Section: Introductionmentioning

confidence: 99%

Predictive Reduced Order Modeling of Chaotic Multi-scale Problems Using Adaptively Sampled Projections

Huang¹,

Duraisamy²

2023

Preprint

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An adaptive projection-based reduced-order model (ROM) formulation is presented for model-order reduction of problems featuring chaotic and convection-dominant physics. An efficient method is formulated to adapt the basis at every time-step of the on-line execution to account for the unresolved dynamics. The adaptive ROM is formulated in a Least-Squares setting using a variable transformation to promote stability and robustness. An efficient strategy is developed to incorporate non-local information in the basis adaptation, significantly enhancing the predictive capabilities of the resulting ROMs. A detailed analysis of the computational complexity is presented, and validated. The adaptive ROM formulation is shown to require negligible offline training and naturally enables both future-state and parametric predictions. The formulation is evaluated on representative reacting flow benchmark problems, demonstrating that the ROMs are capable of providing efficient and accurate predictions including those involving significant changes in dynamics due to parametric variations, and transient phenomena. A key contribution of this work is the development and demonstration of a comprehensive ROM formulation that targets predictive capability in chaotic, multi-scale, and transport-dominated problems.

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Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

Cited by 11 publications

References 62 publications

Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Accelerated solutions of convection‐dominated partial differential equations using implicit feature tracking and empirical quadrature

Numerical Analysis of the Main Wave Propagation Characteristics in a Steel-CFRP Laminate Including Model Order Reduction

Predictive Reduced Order Modeling of Chaotic Multi-scale Problems Using Adaptively Sampled Projections

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