Hierarchical modeling of heterogeneous bodies

Zohdi, Tarek I.; Oden, J. T.; Rodin, Gregory J.

doi:10.1016/s0045-7825(96)01106-1

Cited by 159 publications

(84 citation statements)

References 5 publications

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“…Reliable finite element model simulations of laser therapies can be computed in a fraction of the time that the actual therapy takes place. These prediction capabilities combined with an understanding of HSP kinetics and damage mechanisms at the cellular and tissue levels due to thermal stress, together with recent advances for controlling discretization and modeling errors through adaptive control strategies combined with the modern methods of inverse analysis, calibration, uncertainty quantification, sensitivity analysis, and optimization [12][13][14][15][16][17][18][19][20][21] provide a powerful methodology for planning and optimizing the delivery of thermo-therapy for cancer treatments. Moreover, from a computational point of view, changing the thermal delivery modality merely amounts to changing the source term in the governing PDE.…”

Section: Discussionmentioning

confidence: 99%

Dynamic data‐driven finite element models for laser treatment of cancer

Oden

Diller

Bajaj

et al. 2007

Numerical Methods Partial

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Elevating the temperature of cancerous cells is known to increase their susceptibility to subsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher intensity heat sources may be used to ablate the tissue. These facts are the basis for hyperthermia based cancer treatments. Of the many available modalities for delivering the heat source, the application of a laser heat source under the guidance of real-time treatment data has the potential to provide unprecedented control over the outcome of the treatment process (McNichols et al., Lasers Surg Med 34 (2004), 48-55; Salomir et al., Magn Reson Med 43 (2000), 342-347). The goals of this work are to provide a precise mathematical framework for the real-time finite element solution of the problems of calibration, optimal heat source control, and goal-oriented error estimation applied to the equations of bioheat transfer and demonstrate that current finite element technology, parallel computer architecture, data transfer infrastructure, and thermal imaging modalities are capable of inducing a precise computer controlled temperature field within the biological domain.

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Section: Discussionmentioning

confidence: 99%

Dynamic data‐driven finite element models for laser treatment of cancer

Oden

Diller

Bajaj

et al. 2007

Numerical Methods Partial

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“…These methods are typically based on decomposing the macroscale problem and selective usage of computational techniques discussed in Refs. [5] and [376][377][378], employing a database to directly map the effective behavior from macroscopic information addressed in Refs. [379][380][381][382][383], transformation field analysis [384][385][386][387][388][389][390], or proper orthogonal/generalized decomposition [391-397].…”

Section: Analysis At the Rve Levelmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

190

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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“…ror estimates (indicators of a distance between the macroscopic response and the unknown response of the underlying microscopic structure) [1,[31][32][33][34][35]. Oden and Zohdi [36] developed an adaptive concurrent multiscale method for linear elastic problem with a posteriori error estimator for homogenisation.…”

Section: Introductionmentioning

confidence: 99%

“…In a special case, the error bound for the domain decomposition problem reduces to bounds on effective material properties which was proved to be identical to the Reuss-Voigt inequalities. The work was an extension to their previous paper [31] in which the fine scale discretisation error was ignored. The natural error between the exact solution u and the coarsest scale solution u (0,h) is defined by [43,44]:…”

Section: Introductionmentioning

confidence: 99%

Scale selection in nonlinear fracture mechanics of heterogeneous materials

Akbari

Kerfriden

Bordas

2015

Philosophical Magazine

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A new adaptive multiscale method for the nonlinear fracture simulation of heterogeneous materials is proposed. The two major sources of error in the finite element simulation are discretisation and modelling errors. In the failure problems, the discretisation error increases due to the strain localisation which is also a source for the error in the homogenisation of the underlying micro-structure. In this paper, the discretisation error is controlled by an adaptive mesh refinement procedure following the Zienkiewicz-Zhu technique, and the modelling error, which is the resultant of homogenisation of micro-structure, is controlled by replacing the macroscopic model with the underlying heterogeneous micro-structure. The scale adaptation criterion which is based on an error indicator for homogenisation proposed by[1] is employed for our nonlinear fracture problem. The control of both discretisation and homogenisation errors is the main feature of the proposed multiscale method.

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Hierarchical modeling of heterogeneous bodies

Cited by 159 publications

References 5 publications

Dynamic data‐driven finite element models for laser treatment of cancer

Dynamic data‐driven finite element models for laser treatment of cancer

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Scale selection in nonlinear fracture mechanics of heterogeneous materials

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