Lie-group interpolation and variational recovery for internal variables

Mota, Alejandro; Sun, WaiChing; Ostien, Jakob T.; Foulk, James W.; Long, Kevin Nicholas

doi:10.1007/s00466-013-0876-1

Cited by 44 publications

(49 citation statements)

References 33 publications

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“…Examples of nodal projections include the patch recovery methods and the global L 2 methods of Ortiz and Quigley and Mota et al . .…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

“…Our proposed nodal projection method builds upon the work by Mota et al . , but the distinguishing feature is that a lumped approximation of the projection matrix is utilized in order to minimize the cost of the calculations. Let the globally smooth field for the rotation tensor

\overset{bold-italicZ}{true}

be interpolated using the same Lagrange shape functions as used for the deformation field ϕ h

{\overset{Z}{true}}_{i j} (bold-italicX) = {falsefalse}_{\sum}^{A = 1} N^{A} (bold-italicX) {\overset{Z}{true}}_{italic i j}^{A},

where n u m n p is the number of nodes in the mesh, and N A are the shape functions appearing in the matrix N within the discrete form .…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

“…Let the globally smooth field for the rotation tensor

\overset{bold-italicZ}{true}

be interpolated using the same Lagrange shape functions as used for the deformation field ϕ h

{\overset{Z}{true}}_{i j} (bold-italicX) = {falsefalse}_{\sum}^{A = 1} N^{A} (bold-italicX) {\overset{Z}{true}}_{italic i j}^{A},

where n u m n p is the number of nodes in the mesh, and N A are the shape functions appearing in the matrix N within the discrete form . Then the L 2 projection of the integration point values Z a onto the field

\overset{bold-italicZ}{true}

is expressed through the following integral equation (posed in the reference configuration) : find

\overset{bold-italicZ}{true} \in {scriptV}^{h}

such that for all

bold-italicζ \in {scriptV}^{h}

\int_{normalΩ} bold-italicζ : ((), \overset{bold-italicZ}{true} - bold-italicZ) normald V = 0

with the discrete functional space

{scriptV}^{h} \subset scriptV

defined in terms of the Sobolev space H 1 (Ω) of functions with square‐integrable derivatives

{scriptV}^{h} \subset scriptV = {([], H^{1} (normalΩ))}^{n_{sd} \times n_{sd}}

…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

“…Recently, a framework was proposed by Mota et al . for projections that preserve tensorial properties by invoking the exponential and logarithmic mapping of tensors. Their method represents an enhancement to classical L 2 projection techniques for transferring state variables at integration points between adaptively refined meshes; see, for example, the work of Ortiz and Quigley for strain localization modeling.…”

Section: Introductionmentioning

confidence: 99%

“…Two projection methods are compared: an elemental method and a recently proposed nodal method . The nodal method incorporates the Lie group – Lie algebra relations from to ensure that the projected elastic rotation tensor remains orthogonal. A lumped approximation of the global projection matrix is utilized such that the method is computationally more economical than a mixed formulation.…”

Section: Introductionmentioning

confidence: 99%

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Variational projection methods for gradient crystal plasticity using Lie algebras

Truster

Nassif

2016

Numerical Meth Engineering

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Summary A computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation. While such gradient terms reproduce the size effects exhibited in experiments, incorporating derivatives of the plastic strain yields a nonlocal constitutive model. Rather than applying mixed methods, we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field. Crucially, the projection utilizes the mapping between Lie groups and algebras in order to preserve essential physical properties, such as orthogonality of the plastic rotation tensor. Following the projection, the plastic strain field is directly differentiated to yield the Nye tensor. Additionally, an augmentation scheme is introduced within the global Newton iteration loop such that the computed Nye tensor field is fed back into the stress update procedure. Effectively, this method results in a fully implicit evolution of the constitutive model within a traditional displacement‐based formulation. An elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference. A series of numerical tests are performed for several element types in order to assess the robustness of the method, with emphasis placed upon polycrystalline domains and multi‐axis loading. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Examples of nodal projections include the patch recovery methods and the global L 2 methods of Ortiz and Quigley and Mota et al . .…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

\overset{bold-italicZ}{true}

be interpolated using the same Lagrange shape functions as used for the deformation field ϕ h

{\overset{Z}{true}}_{i j} (bold-italicX) = {falsefalse}_{\sum}^{A = 1} N^{A} (bold-italicX) {\overset{Z}{true}}_{italic i j}^{A},

where n u m n p is the number of nodes in the mesh, and N A are the shape functions appearing in the matrix N within the discrete form .…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

“…Let the globally smooth field for the rotation tensor

\overset{bold-italicZ}{true}

be interpolated using the same Lagrange shape functions as used for the deformation field ϕ h

{\overset{Z}{true}}_{i j} (bold-italicX) = {falsefalse}_{\sum}^{A = 1} N^{A} (bold-italicX) {\overset{Z}{true}}_{italic i j}^{A},

\overset{bold-italicZ}{true}

is expressed through the following integral equation (posed in the reference configuration) : find

\overset{bold-italicZ}{true} \in {scriptV}^{h}

such that for all

bold-italicζ \in {scriptV}^{h}

\int_{normalΩ} bold-italicζ : ((), \overset{bold-italicZ}{true} - bold-italicZ) normald V = 0

with the discrete functional space

{scriptV}^{h} \subset scriptV

defined in terms of the Sobolev space H 1 (Ω) of functions with square‐integrable derivatives

{scriptV}^{h} \subset scriptV = {([], H^{1} (normalΩ))}^{n_{sd} \times n_{sd}}

…”

Section: Spatial Approaches For Evaluating the Nye Tensormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Variational projection methods for gradient crystal plasticity using Lie algebras

Truster

Nassif

2016

Numerical Meth Engineering

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A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain

Sun

Ostien

Salinger

2013

Num Anal Meth Geomechanics

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113

144

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An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro-mechanical problem in fluid-infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple-locking phenomena related to shear failure, incompressibility imposed by pore fluid, and/or incompressible solid skeleton and produce solutions that satisfy the inf-sup condition. The template-based generic programming and automatic differentiation (AD) techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model. Darcy's velocity [7], by using inf-sup stable finite elements (e.g., Talyor-Hood, Raviart-Thomas finite elements) [9,[11][12][13][14], or by applying stabilization procedures to cure the otherwise unstable finite elements [7,[15][16][17][18]. The displacement-Darcy-velocity coupling scheme is relatively easy to implement. However, computation time will be significantly increased because of the extra degrees of freedom added for the pore fluid velocity. Implementing inf-sup stable displacement-pressure finite element requires only one extra degree of freedom for the pore pressure. However, inf-sup stable finite elements require special meshing, data structure, and preprocessing and postprocessing tools to accommodate the need to have different basis functions for the displacement and pore pressure solutions. This difficulty is accompanied with a significant increase in computation cost, as higher-order mixed finite elements lead to larger systems of equations and require more integration points to perform numerical integration, as pointed out in [14].An equal-order displacement-pore pressure mixed finite element formulation is computationally more efficient and does not require significant modification to data structures (except adding an additional degree of freedom to all nodes), and it is therefore more feasible for both code development and maintenance. The price for using the equal-order mixed finite element method, however, is that the mixed finite element formulation must be stabilized by either adding additional terms or introducing enhanced shape functions to eliminate the spurious mode encountered as a result of a failure to satisfy the inf-sup condition. In recent years, effort has been invested to develop stabilization procedures for poromechanics problem under the small strain assumption, for example, [7,[16][17][18]. However, there are numerous occasions in which porous solids experience significant deformation such that a finite strain formulation becomes essential. Examples include water-saturated soil near critical state [7] and hydrated biological tissue during normal physiological activities [19]. In those situations, a stabilized formulation for large deformation poromechanics inheriting the same ...

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Capturing the two‐way hydromechanical coupling effect on fluid‐driven fracture in a dual‐graph lattice beam model

Ulven

Sun

2017

Num Anal Meth Geomechanics

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Fluid-driven fractures of brittle rock is simulated via a dual-graph lattice model. The new discrete hydromechanical model incorporates a two-way coupling mechanism between the discrete element model and the flow network. By adopting an operator-split algorithm, the coupling model is able to replicate the transient poroelasticity coupling mechanism and the resultant Mandel-Cryer hydromechanical coupling effect in a discrete mechanics framework. As crack propagation, coalescence and branching are all path-dependent and irreversible processes, capturing this transient coupling effect is important for capturing the essence of the fluid-driven fracture in simulations. Injection simulations indicate that the onset and propagation of fractures is highly sensitive to the ratio between the injection rate and the effective permeability. Furthermore, we show that in a permeable rock, the borehole breakdown pressure, the pressure at which fractures start to grow from the borehole, depends on both the given ratio between injection rate and permeability and the Biot coefficient.

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Lie-group interpolation and variational recovery for internal variables

Cited by 44 publications

References 33 publications

Variational projection methods for gradient crystal plasticity using Lie algebras

Variational projection methods for gradient crystal plasticity using Lie algebras

A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain

Capturing the two‐way hydromechanical coupling effect on fluid‐driven fracture in a dual‐graph lattice beam model

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