Julián A. Norato

A Computational and Cellular Solids Approach to the Stiffness-Based Design of Bone Scaffolds

We derive a cellular solids approach to the design of bone scaffolds for stiffness and pore size. Specifically, we focus on scaffolds made of stacked, alternating, orthogonal layers of hydroxyapatite rods, such as those obtained via micro-robotic deposition, and aim to determine the rod diameter, spacing and overlap required to obtain specified elastic moduli and pore size. To validate and calibrate the cellular solids model, we employ a finite element model and determine the effective scaffold moduli via numerical homogenization. In order to perform an efficient, automated execution of the numerical studies, we employ a geometry projection method so that analyses corresponding to different scaffold dimensions can be performed on a fixed, non-conforming mesh. Based on the developed model, we provide design charts to aid in the selection of rod diameter, spacing and overlap to be used in the robotic deposition to attain desired elastic moduli and pore size.

Analytical Target Cascading for Multi-Mode Design Optimization: An Engine Case Study

Advances in design and information science have enabled the engineering community to look into changeable systems that work under multiple operating scenarios or modes. In this paper, a multidisciplinary design optimization approach for changeable systems is presented, with its focus on sharing a uniform part of system configuration across all the operating scenarios. Compared to the fully adaptive system approach, this approach enables reduction in the computational expense due to the repetitive mode-by-mode optimization, which becomes impractical as the number of modes increases. In the proposed approach, Analytical Target Cascading (ATC), a hierarchical optimization methodology, models the multi-mode design optimization in a two-level structure: the subsystem problems achieve the performance targets through optimizing local copies of the system configuration; and the system problem coordinates system configuration copies at multiple modes to obtain consistency. Local objectives are introduced to accommodate (unattainable) targets assigned locally for the individual systems, and a weight-updating scheme utilizing local objective information is proposed to balance among performance deviations at multiple modes. A case study on industrial engine simulation parameter identification demonstrates the effectiveness of the proposed approach.

Stress minimization using the level set topology optimization

This paper presents a level set topology optimization method for stress-based structural design problems. Herein, a new sensitivity analysis for a p-norm stress measure function is developed and a level set topology optimization method is used. The methodology is applied to optimize a benchmark L-beam problem with and without holes in the starting design.

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples. [DOI: 10.1115/1.4040624]

Computational design of curvilinear bone scaffolds fabricated via direct ink writing

Abstract Bone scaffold porosity and stiffness play a critical role in the success of critical-size bone defect rehabilitation. In this work, we present a computational procedure to design ceramic bone scaffolds to provide adequate mechanical support and foster bone healing. The scaffolds considered in our study consist of a lattice of curved rods fabricated via direct ink writing. We develop cellular solids models of the scaffold’s effective elastic constants as functions of its geometric parameters, up to some unknown coefficients. To determine numeric values for these coefficients, we execute a computational design of experiments whereby effective elastic properties are obtained using numerical homogenization with the finite element method. In order to automate these experiments and circumvent re-meshing for every scaffold geometry, we project a representative volume element of the scaffold onto a fixed uniform mesh and assign an ersatz material for the analysis. We use these calibrated models in conjunction with finite element analysis and efficient gradient-based optimization methods to design patient-specific scaffolds (i.e., shape, location, and loading) by varying geometric parameters that can be controlled in the fabrication, namely the separation between rods in the lattice, and the printing path of the rods. At present, our methodology is restricted to 2-d idealizations of flat bones subject to in-plane loading. The optimization procedure renders element-wise values of these parameters. As this representation is not amenable to fabrication, lastly, we generate the final scaffold geometry by posing a differential equation whose solution is a function such that its level set lines at specified values correspond to the directrices of the rods in the scaffold. We present examples where we perform the maximization of stiffness of a scaffold implant with a constraint on porosity.