Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

Oñate, Eugenio

doi:10.1186/s40323-016-0065-9

Cited by 5 publications

(3 citation statements)

References 42 publications

(80 reference statements)

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“…For solving stiff ADR equations, various versions of the finite element method (FEM), discontinuous Galerkin method (DGM), finite difference method (FDM) and finite volume method (FVM) have been presented in literature (Huyakorn, 1978; Hughes et al ., 1989, 1998; Badia and Hierro, 2014; Eugenio, 2016; Badia and Bonilla, 2017; Kuzmin, 2009; Lohmann et al , 2017; Fenga et al , 2019; Chen and Shu, 2017; Dong et al , 2017; Uzunca et al , 2014; Anderson et al , 2017; Zeytinoglu et al , 2018; Zhao et al , 2019; Kumar and Chandrashekar, 2019; Wang et al , 2018). Among these high-capacity numerical solvers, finite element-based techniques have been dominantly applied for these highly challenging problems (Huyakorn, 1978; Hughes et al ., 1989, 1998).…”

Section: Introductionmentioning

confidence: 99%

“…The FEMs take advantage of geometric flexibility, weak formulation and the theory of interpolation. Even the classical Galerkin FEM suffers from instability for advection-dominated problems, some stabilization methods together with Galerkin FEM offer better numerical results (Badia and Hierro, 2014; Eugenio, 2016; Badia and Bonilla, 2017; Kuzmin, 2009; Lohmann et al , 2017; Fenga et al , 2019). Despite all these considerable advantages, the p− refinement procedure of the FEM increases the degrees of freedom and computational cost.…”

Section: Introductionmentioning

confidence: 99%

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An implicit-explicit local method for parabolic partial differential equations

Tunc

Sarı

2021

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PurposeThe purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.FindingsThe IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.Originality/valueThe IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An implicit-explicit local method for parabolic partial differential equations

Tunc

Sarı

2021

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“…The FIC approach is based on the incremental solution of a modified system of non-local governing equations accounting for higher order terms obtained by applying the balance laws in domains of finite size. The FIC-based stabilization has been applied in conjunction with the FEM to convection-diffusion and incompressible flows, and solid mechanics [58,141,142]. In those cases, where the convective term has an important role, a first order FIC term is enough to provide stability to the system.…”

Section: Stabilized Formulations For the Shallow Water Equationsmentioning

confidence: 99%

Coupling shallow water models with three-dimensional models for the study of fluid-structure interaction problems using the particle finite element method

Maso Sotomayor

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(English) This thesis investigates numerical methods for the simulation of surface water flows, focusing on the interaction between the large scale and the local scale and its application to natural hazards. Several families of numerical methods for the approximation of large scale phenomena and the coupling with the local scale have been analyzed. The general motion of a fluid mass is governed by the Navier-Stokes equations, which can accurately solve the local scale phenomena. However, the same level of accuracy is not required by the large scale solution of the water-related events. In this context, the shallow water equations are defined. In contrast to the extensive use of the Finite Element Method for solving the Navier-Stokes equations, the shallow-water equations are usually solved with the Finite Volume Method. Thus, an effort have been done to solve both equations in an unified framework. The first part of this thesis is devoted to study stabilized formulations of Finite Element Method for the different forms of the shallow water equations. Stabilized formulations arise from the need to mitigate the various instabilities inherent in numerical approximations. The first source of instability is the incompatibility of the equal interpolation of the variables. The second source of instability is the presence of shocks due to the change of regime or hydraulic jumps. Finally, Gibbs oscillations may appear on the moving shoreline and monotonic properties of the physical system are lost by the numerical approximation. The second part of the thesis is committed to the coupling strategies of the numerical methods for the Navier-Stokes and the shallow water equations. The case of a coupling from the local scale to the large scale is analyzed. This type of coupling corresponds to the generation of cascading natural hazard. The proposed strategy combines a Lagrangian Navier Stokes multi-fluid solver with an Eulerian method based on the Boussinesq equations, an extension of the shallow water equations. Finally, the proposed technique is applied to the numerical simulation of landslide-generated impulse waves. The Particle Finite Element Method has been used to model the landslide runout, its impact against the water body and the consequent wave generation. The results of this fully-resolved analysis are stored at selected interfaces and then used as input for the modelling of waves propagation on the far-field. This one-way coupling scheme drastically reduces the computational cost of the analyses while maintaining high accuracy in reproducing the key phenomena of cascading natural hazards. (Español) En esta tesis se investigan métodos numéricos para la simulación de flujos de aguas superficiales, haciendo énfasis en la interacción entre las distintas escalas y su aplicación a desastres naturales. Se han analizado diversas familias de métodos numéricos para aproximar los fenómenos a gran escala y su acoplamiento con la escala local. El movimiento general de una masa de fluido se rige por las ecuaciones de Navier-Stokes, que pueden resolver con precisión los fenómenos a escala local. Sin embargo, la solución numérica a gran escala de dichos fenómenos, no requiere el mismo nivel de precisión. En este ámbito, se definen las acuaciones de agua poco profundas. En contraste con el amplio uso del Método de los Elementos Finitos para aproximar las ecuaciones de Navier-Stokes, las ecuaciones de aguas poco profundas se suelen resolver con el Método de los Volúmenes Finitos. Por ello, se ha realizado un esfuerzo para resolver ambas ecuaciones en un marco unificado. La primera parte de esta tesis está dedicada a estudiar formulaciones estabilizadas para el Método de los Elementos Finitos aplicado a las diferentes formas de las ecuaciones de aguas someras. Las formulaciones estabilizadas surgen de la necesidad de mitigar las diferentes inestabilidades inherentes a las aproximaciones numéricas. La primera fuente de inestabilidad es la incompatibilidad debida a la interpolación de las variables. La segunda fuente de inestabilidad es la presencia de discontinuidades debidos al cambio de régimen o a los saltos hidráulicos. Por último, pueden aparecer oscilaciones de Gibbs en la línea de costa en movimiento, dado que las propiedades monótonas del sistema físico se pierden por la aproximación numérica. La segunda parte de la tesis está dedicada a las estrategias de acoplamiento de los métodos numéricos para las ecuaciones de Navier-Stokes y de aguas poco profundas. Se ha analizado el caso de acoplamiento desde la escala local a la escala global. Este tipo de acoplamiento corresponde a la generación de desastres naturales en cascada. La estrategia propuesta combina un solver Lagrangiano de Navier Stokes para multi-fluidos con un método Euleriano basado en las ecuaciones de Boussinesq, una extensión de las ecuaciones de aguas someras. Finalmente, la técnica propuesta se ha aplicado a la simulación numérica de olas generadas por deslizamientos. El deslizamiento de ladera, su impacto contra la masa de agua y la consiguiente generación de olas se ha modelado con el Método de Elementos Finitios de Partículas. Los resultados de este análisis detallado se almacenan en las interfaces seleccionadas que, luego, se utilizan como punto de entrada para modelar la propagación de olas en el campo lejano. Este esquema de acoplamiento unidireccional reduce drásticamente el coste computacional, a la vez que se mantiene una alta precisión en la simulación de los fenómenos clave de desastres naturales.

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Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants

Felippa

Oñate

Idelsohn

2017

Arch Computat Methods Eng

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This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). That formulation utilizes high order derivatives of the classical fields of continuum mechanics integrated over control regions to construct stabilizing modification terms. Fields include displacements, body forces, strains, stresses, pressure and volumetric strains. To support observerinvariant FIC formulations, we have catalogued field transformation equations as well as sets of linear and quadratic invariants of fields and of their derivatives up to appropriate order. Attention is focused on the two-dimensional case of a body in plane strain under the drilling-rotation transformation group. Results are presented for displacement and body-force derivatives of orders up to 4, and for stress, strain, pressure and volumetric strain derivatives of order up to 3. The material assembled here is self-contained because this catalog is believed to be useful for non-FIC applications; for example gradient-based, nonlocal constitutive models of multiscale mechanics and physics that involve finite characteristic dimensions analogous to FIC steplengths.

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Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

Cited by 5 publications

References 42 publications

An implicit-explicit local method for parabolic partial differential equations

An implicit-explicit local method for parabolic partial differential equations

Coupling shallow water models with three-dimensional models for the study of fluid-structure interaction problems using the particle finite element method

Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants

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