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Fundamental Relationships for Flow and Transport
Published in James L. Martin, Steven C. McCutcheon, Robert W. Schottman, Hydrodynamics and Transport for Water Quality Modeling, 2018
James L. Martin, Steven C. McCutcheon, Robert W. Schottman
which is known as a forward-difference approximation. But note the introduction of a source of error by neglecting the higher-order terms. The approximation error introduced by neglecting the higher-order terms is truncation error.
Discrete empirical interpolation and unfitted mesh FEMs: application in PDE-constrained optimization
Published in Optimization, 2023
Georgios Katsouleas, Efthymios N. Karatzas, Fotios Travlopanos
In the following, we provide details regarding the computational performance of the method for the test case of the randomly chosen parameter value . Regrading DEIM approximations to the non-affine components of the OCP system, the relevant -errors have been computed as follows: DEIM approximation -error for : ,DEIM approximation -error for : ,DEIM approximation -error for : ,DEIM approximation -error for : .
Lightweight topology optimization of thermal structures under compliance, stress and temperature constraints
Published in Journal of Thermal Stresses, 2021
Qingxuan Meng, Bin Xu, Chenguang Huang, Ge Wang
Since the maximum stress of structure needs to meet the stress limit in thermo-elastic TO problems,it is necessary to tackle a great number of stress constraints due to the fact that stress is a local quantity. This issue would make optimization process too expensive for computational cost. To address the difficulty, the local stress metrics are replaced with a few global stress metrics. Here, P-norm function is adopted through normalization and it is written as where P means the stress aggregation parameter. However, there is the approximation error between the P-norm function and the maximum stress of structure. It should be emphasized that P has an obvious effect on the approximation error and the degree of smoothness. When P tends to be infinite, equals to because the approximation error is negligible. Although a larger value of P can improve the solution quality of TO problems, it could also cause higher non-linearity for the P-norm function, which may bring iterative oscillation and convergence difficulty. Therefore, it is necessary to develop effective schemes to solve above problems.
H2 model reduction for negative imaginary systems
Published in International Journal of Control, 2020
If the approximation error system (3) is unstable, its norm is defined to be . Otherwise, the norm of is defined as (Gugercin et al., 2008) In this paper, we are interested in finding a reduced-order stable NI system that minimises the norm of the approximation error system . Hence, the model reduction cost function can be written as Let denote the set of the reduced-order stable NI systems with the form (2). Given a stable NI system (1) and , the model reduction problem for NI systems can be formulated as the minimisation problem