Colby C. Swan

Voigt–Reuss topology optimization for structures with linear elastic material behaviours

The desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems. To achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical Reuss (compliant) and Voigt (sti) mixing rules is investigated. To avoid checkerboarding instabilities, the continuous topology optimization formulation is coupled with a novel spatial ltering procedure. The issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the sti Voigt formulation to the compliant Reuss formulation. The very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature. ? 1997 by John Wiley & Sons, Ltd.

VOIGT-REUSS TOPOLOGY OPTIMIZATION FOR STRUCTURES WITH NONLINEAR MATERIAL BEHAVIORS

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance.

Micromechanically Based Poroelastic Modeling of Fluid Flow in Haversian Bone

To explore the hypothesis that load-induced fluid flow in bone is a mechano-transduction mechanism in bone adaptation, unit cell micro-mechanical techniques are used to relate the microstructure of Haversian cortical bone to its effective poroelastic properties. Computational poroelastic models are then applied to compute in vitro Haversian fluid flows in a prismatic specimen of cortical bone during harmonic bending excitations over the frequency range of 10(0) to 10(6) Hz. At each frequency considered, the steady state harmonic response of the poroelastic bone specimen is computed using complex frequency-domain finite element analysis. At the higher frequencies considered, the breakdown of Poisueille flow in Haversian canals is modeled by introduction of a complex fluid viscosity. Peak bone fluid pressures are found to increase linearly with loading frequency in proportion to peak bone stress up to frequencies of approximately 10 kHz. Haversian fluid shear stresses are found to increase linearly with excitation frequency and loading magnitude up until the breakdown of Poisueille flow. Tan delta values associated with the energy dissipated by load-induced fluid flow are also compared with values measured experimentally in a concurrent broadband spectral analysis of bone. The computational models indicate that fluid shear stresses and fluid pressures in the Haversian system could, under physiologically realistic loading, easily reach the level of a few Pascals, which have been shown in other works to elicit cell responses in vitro.

Limit state analysis of earthen slopes using dual continuum/FEM approaches

A framework alternative to that of classical slope stability analysis is developed, wherein the soil mass is treated as a continuum and in-situ soil stresses and strengths are computed accurately using inelastic finite element methods with general constitutive models. Within this framework, two alternative methods of stability analysis are presented. In the first, the strength characteristics of the soil mass are held constant, and the gravitational loading on the slope system is increased until failure is initiated by well-defined mechanisms. In the second approach, the gravity loading on the slope system is held constant, while the strength parameters of the soil mass are gradually decreased until well-defined failure mechanisms develop. Details on applying both of the proposed methods, and comparisons of their characteristics on a number of solved example problems are presented.

A symmetry reduction method for continuum structural topology optimization

Abstract It is considered that asymmetrical material layout design solutions are caused by numerical roundoff and the convexity characteristics of alternative topology design formulations. Emphasis is placed here not on analyzing potential instabilities that lead to asymmetrical designs, but on a method to stabilize topology design formulations. A novel symmetry reduction method is proposed, implemented and studied. While enforcing symmetry and significantly reducing the size of the optimization problem, the symmetry reduction method is shown to have the added benefit of greatly simplified design sensitivity analysis of non-simple repeated vibrational eigenvalues which occur in many symmetrical structures.

Modeling Deformation-Induced Fluid Flow in Cortical Bone's Canalicular-Lacunar System

To explore the potential role that load-induced fluid flow plays as a mechano–transduction mechanism in bone adaptation, a lacunar–canalicular scale bone poroelasticity model is developed and implemented. The model uses micromechanics to homogenize the pericanalicular bone matrix, a system of straight circular cylinders in the bone matrix through which bone fluids can flow, as a locally anisotropic poroelastic medium. In this work, a simplified two-dimensional model of a periodic array of lacunae and their surrounding systems of canaliculi is used to quantify local fluid flow characteristics in the vicinity of a single lacuna. When the cortical bone model is loaded, microscale stress, and strain concentrations occur in the vicinity of individual lacunae and give rise to microscale spatial variations in the pore fluid pressure field. Furthermore, loading of the bone matrix containing canaliculi generates fluid pressures in the contained fluids. Consequently, loading of cortical bone induces fluid flow in the canaliculi and exchange of fluid between canaliculi and lacunae. For realistic bone morphology parameters, and a range of loading frequencies, fluid pressures and fluid–solid drag forces in the canalicular bone are computed and the associated energy dissipation in the models compared to that measured in physical in vitro experiments on human cortical bone. The proposed model indicates that deformation-induced fluid pressures in the lacunar–canalicular system have relaxation times on the order of milliseconds as opposed to the much shorter times (hundredths of milliseconds) associated with deformation-induced pressures in the Haversian system.

A broadband viscoelastic spectroscopic study of bovine bone: implications for fluid flow.

AbstractTo explore the hypothesis that mechanical excitation-induced fluid flow and/or fluid pressure are potential mechanical transduction mechanisms in bone adaptation, a complementary experimental and analytical modeling effort has been undertaken. Experimentally, viscoelastic tanδ properties of saturated cortical bovine bone were measured in both torsion and bending, and significant tan δ values in the 100-105Hz range were observed, although the nature of the damping is not consistent with a fluid pressure hypothesis. Analytically, micromechanically based poroelasticity models were exercised to quantify energy dissipation associated with load-induced fluid flow in large scale channels. The modeling results indicate that significant damping due to fluid flow occurs only above 1 MHz frequencies. Together, the experimental and analytical results indicate that at excitation frequencies presumed to be physiological (1–100 Hz), mechanical loading of bone generates extremely small pore fluid pressures, making the hypothesized fluid-pressure transduction mechanism upon osteocytes untenable.

CONTINUUM TOPOLOGY OPTIMIZATION OF BUCKLING-SENSITIVE STRUCTURES

Twoformulationsforcontinuum topologyoptimizationofstructurestakingbucklingconsiderationsinto account are developed, implemented, and compared. In thee rst, the structure undergoing a specie ed loading is modeled as a hyperelastic continuum at e nite deformations and is optimized to maximize the minimum critical buckling load. In the second, the structure under a similar loading is modeled as linear elastic, and the critical buckling load is computed with linearized buckling analysis. Specie c issues addressed include usage of suitable “ mixing rules,” a node-based design variable formulation, techniques for eliminating regions devoid of structural material from the analysis problem, and consistent design sensitivity analysis. The performance of the formulations is demonstrated on the design of different structures. When problems are solved with moderate loads and generous material usage constraints, designs using compression and tension members are realized. Alternatively, when fairly large loads together with very stringent material usage constraints are imposed, structures utilizing primarily tension members result. Issues that arise when designing very light structures with stringent material usage constraints are discussed along with the importance of considering potential geometrical instabilities in the concept design of structural systems.

Homogenization-based analysis and design of composites

Abstract Computational homogenization is demonstrated as a potent analysis tool that can be used directly to predict the property-structure relationships of many existing classes of composites, and indirectly to design the topological macrostructure of new generations of composites so as to optimize their mechanical properties. This paper lays out the homogenization analysis problem for general classes of inelastic mechanical composites. The analysis techniques presented are logically divided into stress- and strain-controlled methods, both of which are formulated for periodic composites in a finite element setting. A well-recognized issue with computational homogenization is that for three-dimensional structures associated computing costs escalate rapidly with mesh refinement, thus providing a potential obstacle to usage of the method. To address this important issue, the relative performance of alternative vector and parallel numerical algorithms that facilitate high speed and efficiency with computing resources are compared on sample homogenization computations of inelastic Byzantine masonry and modern graphite-epoxy. Building upon the established homogenization analysis framework, a novel method for designing the topology of a composites macrostructure is then formulated, implemented and demonstrated to achieve new material designs with significantly enhanced mechanical performance properties.

Dynamic Motion Planning of 3D Human Locomotion Using Gradient-Based Optimization

Since humans can walk with an infinite variety of postures and limb movements, there is no unique solution to the modeling problem to predict human gait motions. Accordingly, we test herein the hypothesis that the redundancy of human walking mechanisms makes solving for human joint profiles and force time histories an indeterminate problem best solved by inverse dynamics and optimization methods. A new optimization-based human-modeling framework is thus described for predicting three-dimensional human gait motions on level and inclined planes. The basic unknowns in the framework are the joint motion time histories of a 25-degree-of-freedom human model and its six global degrees of freedom. The joint motion histories are calculated by minimizing an objective function such as deviation of the trunk from upright posture that relates to the human models performance. A variety of important constraints are imposed on the optimization problem, including (1) satisfaction of dynamic equilibrium equations by requiring the models zero moment point (ZMP) to lie within the instantaneous geometrical base of support, (2) foot collision avoidance, (3) limits on ground-foot friction, and (4) vanishing yawing moment. Analytical forms of objective and constraint functions are presented and discussed for the proposed human-modeling framework in which the resulting optimization problems are solved using gradient-based mathematical programming techniques. When the framework is applied to the modeling of bipedal locomotion on level and inclined planes, acyclic human walking motions that are smooth and realistic as opposed to less natural robotic motions are obtained. The aspects of the modeling framework requiring further investigation and refinement, as well as potential applications of the framework in biomechanics, are discussed.