The Cream in My Polytope

This guided discovery project provides high school math students an opportunity to solve a real-world problem by applying tools they have learned in geometry and, optionally, calculus. The task is to determine the volume, surface area, and other properties of a certain form of folded paper container, such as those used by Wendy’s for sour cream. Students first construct the containers, and then are challenged to find a geometric way to determine the volume of these objects, which do not fit conventional volume formulas. They next find surface area and, optionally, angles of the container. In the process, 2- and 3-dimensional visualization skills are exercised. Calculus students are asked to find the volume by the method of slabs, and subsequently to determine the dimensions that optimize the surface-to-volume ratio. Several additional extensions are suggested, and core curriculum standards are listed. Diagrams and calculations are provided. While much support material is provided, it is hoped that the teacher will encourage students to explore, discover and invent as much as possible on their own

Mike's Problem - A Proposed Solution

Finally!  A proposed solution to "Mike's Problem" presented in the Spring, 2001, edition of OSJM

Efficient two-scale FE-FFT-based mechanical process simulation of elasto-viscoplastic polycrystals at finite strains

The purpose of this work is the development of an efficient two-scale numerical scheme for the prediction of the local and overall mechanical behavior of polycrystalline materials with elasto-viscoplastic constitutive behavior at finite strains. Assuming scale separation, the microstructural deformations are prescribed by the kinematics of the macroscopic continuum body. The macroscopic constitutive behavior is in turn determined by the mean response of the point-wise linked microstructure which is represented by a periodic unit cell. The algorithmic formulation and numerical solution of the two locally coupled boundary value problems is based on the FE-FFT method. In particular, the presented work is concerned with the development of a CPU- and memory-efficient solution strategy for two-scale finite strain crystal plasticity simulations of polycrystalline aggregates which is based on a microstructural convergence analysis. This efficient solution strategy allows a two-scale simulation of complex macroscopic boundary value problems in a reasonable time period. In order to demonstrate the versatile use of the proposed method, three polycrystalline materials namely copper, aluminum and iron are studied with different textures for three distinct macroscopic loading conditions. On this basis, the micromechanical fields and the overall material response of an iron-based polycrystal are predicted for a deep rolling process, which serves as a testing example for a representative and application oriented simulation. © 2020 Elsevier B.V