An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. II: Implementation and Numerical Experiments

Nochetto, Ricardo H.; Paolini, Maurizio; Verdi, C.

doi:10.1137/0912065

Cited by 54 publications

(43 citation statements)

References 16 publications

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“…On the other hand, (6.18) reflects the hyperbolic structure of the interface as well as the parabolic structure of the problem elsewhere. We refer to [11] for a similar observation. where r := (x − 1 3 ) 2 + (y − α(t)) 2 1/2 .…”

Section: Element Indicatorsmentioning

confidence: 59%

Untitled

2000

Math. Comp.

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Abstract. Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

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Section: Element Indicatorsmentioning

confidence: 59%

Untitled

2000

Math. Comp.

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“…We then propose an adaptive algorithm which uses these criteria while simultaneously performing the usual local mesh refinement and equilibration of the spatial and temporal errors. This algorithm is inspired from [34,35,40,36,6] and from the work [18,23,14,19]. We conclude Section 4 by proving that, under these criteria, our estimators are also efficient while representing a lower bound for the dual norm of the residual.…”

Section: Introductionmentioning

confidence: 94%

“…We consider the two-dimensional test case proposed by Nochetto et al [34,35] on the space-time domain Ω × (0, T ) with Ω = (0, 5) 2 and T = π/1.25. The function β(·) is given by β(u) = u ⊖ + (u − 1)…”

Section: Settingmentioning

confidence: 99%

“…Therein, the author derives estimates based on the residual equation for a two-dimensional regularized Stefan problem and proposes a space adaptive finite element algorithm. A posteriori indicators for unsteady phase change problems were derived by Nochetto et al in [34,35], together with an adaptive algorithm which equilibrates space and time discretization errors. Many other adaptive refinement algorithms such as that of Beckett et al [6] have also been proposed.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

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“…the heat source) [121]. Therefore, I use the adaptive mesh approach to make the mesh sizes adaptive to the local physical conditions [122][123][124][125] (see Figure 6 (b)). Once the CAD design file and the printing process settings are given, the G-code [126] can be generated by slicing software, such as Slic3r.…”

Section: Transient Finite Element Analysismentioning

confidence: 99%

Integration of physically-based and data-driven approaches for thermal field prediction in additive manufacturing

Jin

2018

Materials & Design

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A quantitative understanding of thermal field evolution is vital for quality control in additive manufacturing (AM). Because of the unknown material parameters, high computational costs, and imperfect understanding of the underlying science, physically-based approaches alone are insufficient for component-scale thermal field prediction. Here, I present a new framework that integrates physically-based and data-driven approaches with quasi in situ thermal imaging to address this problem. The framework consists of (i) thermal modeling using 3D finite element analysis (FEA), (ii) surrogate modeling using functional Gaussian process, and (iii) Bayesian calibration using the thermal imaging data. Based on heat transfer laws, I first investigate the transient thermal behavior during AM using 3D FEA. A functional Gaussian process-based surrogate model is then constructed to reduce the computational costs from the high-fidelity, physically-based model. I finally employ a Bayesian calibration method, which incorporates the surrogate model and thermal measurements, to enable layer-to-layer thermal field prediction across the whole component. A case study on fused deposition modeling is conducted for components with 7 to 16 layers. The cross-validation results show that the proposed framework allows for accurate and fast thermal field prediction for components with different process settings and geometric designs. Integration of Physically-based and Data-driven Approaches for Thermal Field Prediction in Additive ManufacturingJingran Li GENERAL AUDIENCE ABSTRACT This paper aims to achieve the layer to layer temperature monitoring and consequently predict the temperature distribution for any new freeform geometry. An engineering statistical synergistic model is proposed to integrate the pure statistical methods and finite element modeling (FEM), which is physically meaningful as well as accurate for temperature prediction. Besides, this proposed synergistic model contains geometry information, which can be applied to any freeform geometry. This paper serves to enable a holistic cyber physical systems-based approach for the additive manufacturing (AM) not only restricted in fused deposition modeling (FDM) process but also can be extended to powder-based process like laser engineered net shaping (LENS) and selective laser sintering (SLS). This paper as well as the scheduled future works will make it affordable for customized AM including customized geometries and materials, which will greatly accelerate the transition from rapid prototyping to rapid manufacturing. This article demonstrates a first evaluation of engineering statistical synergistic model in AM technology, which gives a perspective on future researches about online quality monitoring and control of AM based data fusion principles. iv ACKNOWLEDGEMENTS

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An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. II: Implementation and Numerical Experiments

Cited by 54 publications

References 16 publications

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Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Integration of physically-based and data-driven approaches for thermal field prediction in additive manufacturing

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