Jacob Fish

A finite element with embedded localization zones

Abstract A method is developed by which localization zones can be embedded in four-node quadrilaterals and related elements. This is accomplished by modifying the strain field within the framework of a three-field variational statement. The jumps in strain associated with the localization band are obtained by imposing traction continuity and compatibility within the element; the latter follows naturally from the variational statement. Several one- and two-dimensional applications are shown.

Effect of aggregation on thermal conduction in colloidal nanofluids

Using effective medium theory the authors demonstrate that the thermal conductivity of nanofluids can be significantly enhanced by the aggregation of nanoparticles into clusters. Predictions of the effective medium theory are in excellent agreement with detailed numerical calculation on model nanofluids involving fractal clusters and show the importance of cluster morphology on thermal conductivity enhancements.

Role of Brownian motion hydrodynamics on nanofluid thermal conductivity

We use a simple kinetic theory based analysis of heat flow in fluid suspensions of solid nanoparticles (nanofluids) to demonstrate that the hydrodynamics effects associated with Brownian motion have a minor effect on the thermal conductivity of the nanofluid. Our conjecture is supported by the results of molecular dynamics simulations of heat flow in a model nanofluid with well-dispersed particles. Our findings are consistent with the predictions of the effective medium theory as well as with recent experimental results on well dispersed metal nanoparticle suspensions.

Computational plasticity for composite structures based on mathematical homogenization: Theory and practice

This paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains. Starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases. The overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro-constituent, and history data is updated at two points (reinforcement and matrix) in the microstructure, one for each phase. Macroscopic history data is stored in the database and then subjected in the post-processing stage onto the unit cell in the critical locations. For numerical examples considered, the CPU time obtained by means of the two-point averaging scheme with variational micro-history recovery with 30 seconds on SPARC 1051 as opposed to 7 hours using classical mathematical homogenization theory. At the same time the maximum error in the microstress field in the critical unit cell was only 3.5% in comparison with the classical mathematical homogenization theory.

The s-version of the finite element method

Abstract A methodology to improve the quality of the finite element calculations in the regions of unacceptable errors has been developed. Unlike the existing adaptive techniques, where either the mesh is refined ( h -version), or the polynomial order is increased ( p -version), or a combination of both ( h − p version), the s -version increases the resolution by superimposing additional mesh(es) of higher-order hierarchical elements. C 0 continuity of the displacement field is maintained by imposing homogeneous boundary conditions on the superimposed field in the portion of the boundary which is not contained within the boundary of the problem. The superimposed regions can be of arbitrary shape, unlimited by the problem geometry, boundary conditions and the underlying mesh topography. Numerical experiments for linear problems involving singularities and smooth solutions as well as the shear banding problem in viscoplastic solid, are presented to validate the present formulation.

Computational damage mechanics for composite materials based on mathematical homogenization

This paper is aimed at developing a non-local theory for obtaining numerical approximation to a boundary value problem describing damage phenomena in a brittle composite material. The mathematical homogenization method based on double-scale asymptotic expansion is generalized to account for damage effects in heterogeneous media. A closed-form expression relating local fields to the overall strain and damage is derived. Non-local damage theory is developed by introducing the concept of non-local phase fields (stress, strain, free energy density, damage release rate, etc.) in a manner analogous to that currently practiced in concrete [1, 2], with the only exception being that the weight functions are taken to be C0 continuous over a single phase and zero elsewhere. Numerical results of our model were found to be in good agreement with experimental data of 4-point bend test conducted on composite beam made of Blackglas™/Nextel 5-harness satin weave. Copyright

A Dispersive Model for Wave Propagation in Periodic Heterogeneous Media Based on Homogenization With Multiple Spatial and Temporal Scales

A dispersive model is developed for wave propagation in periodic heterogeneous media. The model is based on the higher order mathematical homogenization theory with multiple spatial and temporal scales. A fast spatial scale and a slow temporal scale are introduced to account for the rapid spatial fluctuations as well as to capture the long-term behavior of the homogenized solution. By this approach the problem of secularity, which arises in the conventional multiple-scale higher order homogenization of wave equations with oscillatory coefficients, is successfully resolved. A model initial boundary value problem is analytically solved and the results have been found to be in good agreement with a numerical solution of the source problem in a heterogeneous medium.

Multiscale analysis of composite materials and structures

Abstract The manuscript presents a nonlinear multiscale computational procedure based on the philosophy of multilevel methods and which exploits the similarity between the classical engineering global–local design practice and multilevel methods. Both approaches separate the system response into the global and local effects. While classical substructuring-based schemes separate the global and local effects on the basis of domain decomposition principles often resorting to intuition and understanding the physics of the problem, multilevel approaches, carry out de facto a spectral decomposition automatically as part of the solution process. Seven multiscale problems were considered to validate the present formulation.

Multi-grid method for periodic heterogeneous media Part 2: Multiscale modeling and quality control in multidimensional case

Abstract A multi-grid method for a periodic heterogeneous medium in multidimensions is developed. Based on the homogenization theory, special intergrid transfer operators have been developed to stimulate a low frequency response of the boundary value problem with oscillatory coefficients. An adaptive strategy is developed to form a nearly optimal two-scale computational model consisting of the finite element mesh entirely constructed on the microscale in the regions identified by the idealization error indicators, while elsewhere, the modeling level is only sufficient to capture the response of homogenized medium. Numerical experiments show the usefulness of the proposed adaptive multi-level procedure for predicting a detailed response of composite specimens.

Multiscale finite element method for a locally nonperiodic heterogeneous medium

A generalization of the mathematical homogenization theory to account for locally nonperiodic solutions is presented. Such nonperiodicity may arise either due to the rapidly varying microstructure (e.g.: graded materials, microcracks) or because the macroscopic solution is not smooth and may have significant variation within a microstructure. In the portion of the problem domain where the material is formed by a spatial repetition of the base cell and the macroscopic solution is smooth, a double scale asymptotic expansion and solution periodicity are assumed, and consequently, mathematical homogenization theory is employed to uncouple the microscopic problem from the global solution. For the rest of the problem domain it is assumed that the periodic solution does not exist (cutouts, cracks, free edges in composites, etc.) and the approximation space is decomposed into macroscopic and microscopic fields. Compatibility between the two regions is explicitly enforced. The proposed method is applied to resolve the structure of the microscopic fields in the single ply composite plates with a centered hole and with a centered crack and in the [0/90]s laminated plate. Numerical results are compared to the reference solution, an engineering global-local approach, and the direct extraction from the mathematical homogenization method.