433 research outputs found

    A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces

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    In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.Comment: 28 pages, 16 figures, 3 table

    Inverse designing surface curvatures by deep learning

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    Smooth and curved microstructural topologies found in nature - from soap films to trabecular bone - have inspired several mimetic design spaces for architected metamaterials and bio-scaffolds. However, the design approaches so far have been ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Here, we explore surface curvature as a design modality and present a deep learning framework to produce topologies with as-desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, we demonstrate successful generalization beyond both the design and data space by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio-scaffolds and implants. Lastly, we bridge curvature and mechanics by showing how topological curvature can be designed to promote mechanically beneficial stretching-dominated deformation over bending-dominated deformation.Comment: 23 pages, 12 figure

    Strong-Segregation Theory of Bicontinuous Phases in Block Copolymers

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    We compute phase diagrams for AnBmA_nB_m starblock copolymers in the strong-segregation regime as a function of volume fraction Ď•\phi, including bicontinuous phases related to minimal surfaces (G, D, and P surfaces) as candidate structures. We present the details of a general method to compute free energies in the strong segregation limit, and demonstrate that the gyroid G phase is the most nearly stable among the bicontinuous phases considered. We explore some effects of conformational asymmetry on the topology of the phase diagram.Comment: 14 pages, latex, 21 figures, to appear in Macromolecule

    Minimality via second variation for a nonlocal isoperimetric problem

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    We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L1L^1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.Comment: 35 page

    Modeling, assessment, and design of porous cells based on schwartz primitive surface for bone scaffolds

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    The design of bone scafolds for tissue regeneration is a topic of great interest, which involves diferent issues related to geometry of architectures, mechanical behavior, and biological requirements, whose optimal combination determines the success of an implant. Additive manufacturing (AM) has widened the capability to produce structures with complex geometries, which should potentially satisfy the diferent requirements. These architectures can be obtained by means of refned methods and have to be assessed in terms of geometrical and mechanical properties. In this paper a triply periodic minimal surface (TPMS), the Schwarz's Primitive surface (P-surface), has been considered as scafold unit cell and conveniently parameterized in order to investigate the efect of modulation of analytical parameters on the P-cell geometry and on its properties. Several are the cell properties, which can afect the scafold performance. Due to the important biofunctional role that the surface curvature plays in mechanisms of cellular proliferation and diferentiation, in this paper, in addition to properties considering the cell geometry in its whole (such as volume fraction or pore size), new properties were proposed. Tese properties involve, particularly, the evaluation of local geometrical-diferential properties of the P-surface. Te results of this P-cell comprehensive characterization are very useful for the design of customized bone scafolds able to satisfy both biological and mechanical requirements. A numerical structural evaluation, by means of fnite element method (FEM), was performed in order to assess the stifness of solid P-cells as a function of the changes of the analytical parameters of outer surface and the thickness of cell. Finally, the relationship between stifness and porosity has been analyzed, given the relevance that this property has for bone scafolds design

    Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

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    Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers

    CAD-Based Porous Scaffold Design of Intervertebral Discs in Tissue Engineering

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    With the development and maturity of three-dimensional (3D) printing technology over the past decade, 3D printing has been widely investigated and applied in the field of tissue engineering to repair damaged tissues or organs, such as muscles, skin, and bones, Although a number of automated fabrication methods have been developed to create superior bio-scaffolds with specific surface properties and porosity, the major challenges still focus on how to fabricate 3D natural biodegradable scaffolds that have tailor properties such as intricate architecture, porosity, and interconnectivity in order to provide the needed structural integrity, strength, transport, and ideal microenvironment for cell- and tissue-growth. In this dissertation, a robust pipeline of fabricating bio-functional porous scaffolds of intervertebral discs based on different innovative porous design methodologies is illustrated. Firstly, a triply periodic minimal surface (TPMS) based parameterization method, which has overcome the integrity problem of traditional TPMS method, is presented in Chapter 3. Then, an implicit surface modeling (ISM) approach using tetrahedral implicit surface (TIS) is demonstrated and compared with the TPMS method in Chapter 4. In Chapter 5, we present an advanced porous design method with higher flexibility using anisotropic radial basis function (ARBF) and volumetric meshes. Based on all these advanced porous design methods, the 3D model of a bio-functional porous intervertebral disc scaffold can be easily designed and its physical model can also be manufactured through 3D printing. However, due to the unique shape of each intervertebral disc and the intricate topological relationship between the intervertebral discs and the spine, the accurate localization and segmentation of dysfunctional discs are regarded as another obstacle to fabricating porous 3D disc models. To that end, we discuss in Chapter 6 a segmentation technique of intervertebral discs from CT-scanned medical images by using deep convolutional neural networks. Additionally, some examples of applying different porous designs on the segmented intervertebral disc models are demonstrated in Chapter 6

    Bending Frustration of Lipid-Water Mesophases Based on Cubic Minimal Surfaces

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    Inverse bicontinuous cubic phases are ubiquitous in lipid-water mixtures and consist of a lipid bilayer forming a cubic minimal surface, thereby dividing space into two cubic networks of water channels. For small hydrocarbon chain lengths, the monolayers can be modeled as parallel surfaces to a minimal midsurface. The bending energy of the cubic phases is determined by the distribution of Gaussian curvature over the minimal midsurfaces which we calculate for seven different structures (G, D, P, I-WP, C(P), S and F-RD). We show that the free-energy densities of the structures G, D and P are considerably lower than those of the other investigated structures due to their narrow distribution of Gaussian curvature. The Bonnet transformation between G, D, and P implies that these phases coexist along a triple line, which also includes an excess water phase. Our model includes thermal membrane undulations. Our qualitative predictions remain unchanged when higher order terms in the curvature energy are included. Calculated phase diagrams agree well with the experimental results for 2:1 lauric acid/dilauroyl phosphatidylcholine and water.Comment: Revtex, 23 pages with 9 postscript figures included, to appear in Langmui
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