Optimal and continuous multilattice embedding

Because of increased geometric freedom at a widening range of length scales and access to a growing material space, additive manufacturing has spurred renewed interest in topology optimization of parts with spatially varying material properties and structural hierarchy. Simultaneously, a surge of micro/nanoarchitected materials have been demonstrated. Nevertheless, multiscale design and micro/nanoscale additive manufacturing have yet to be sufficiently integrated to achieve free-form, multiscale, biomimetic structures. We unify design and manufacturing of spatially varying, hierarchical structures through a multimicrostructure topology optimization formulation with continuous multimicrostructure embedding. The approach leads to an optimized layout of multiple microstructural materials within an optimized macrostructure geometry, manufactured with continuously graded interfaces. To make the process modular and controllable and to avoid prohibitively expensive surface representations, we embed the microstructures directly into the 3D printer slices. The ideas provide a critical, interdisciplinary link at the convergence of material and structure in optimal design and manufacturing.

The μ-Basis of Improper Rational Parametric Surface and Its Application

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

Father Iakinf Bičurin’s theory of mental inflection in Chinese

Summary The article deals with the highly original idea of mental inflection [умственное словоизменение] in Chinese put forward by the prominent nineteenth-century Russian sinologist Nikita (比丘林, Father Iakinf or Hyacinthus) Bičurin (1777–1853) in his Kitajskaja Grammatika (1835) and other language-related works. The concept refers to the internal features of Chinese morphology which compensate for the absence of common grammatical inflection. Fostered in an Humboldtian spirit, the theory established a link between covert categories and their surface representations a century before functional syntax appeared on the linguistic stage.

Exploring Novel Surface Representations via an Experimental Ray-Tracer in CGA

Conformal Geometric Algebra (CGA) provides a unified representation of both geometric primitives and conformal transformations, and as such holds significant promise in the field of computer graphics. In this paper we implement a simple ray tracer in CGA with a Blinn–Phong lighting model, before putting it to use to examine ray intersections with surfaces generated from the direct interpolation of geometric primitives. General surfaces formed from these interpolations are rendered using analytic normals. In addition, special cases of point-pair interpolation, which might find use in graphics applications, are described and rendered. A closed form expression is found for the derivative of the square root of a scalar plus 4-vector element with respect to a scalar parameter. This square root derivative is used to construct an expression for the derivative of a pure-grade multivector projected to the blade manifold. The blade manifold projection provides an analytical method for finding the normal line to the interpolated surfaces and its use is shown in lighting calculations for the ray tracer and in generating vertex normals for exporting the evolved surfaces as polygonal meshes.

Detailed 3D Fault Representations for the 2019 Ridgecrest, California, Earthquake Sequence

ABSTRACT We present new 3D source fault representations for the 2019 M 6.4 and M 7.1 Ridgecrest earthquake sequence. These representations are based on relocated hypocenter catalogs expanded by template matching and focal mechanisms for M 4 and larger events. Following the approach of Riesner et al. (2017), we generate reproducible 3D fault geometries by integrating hypocenter, nodal plane, and surface rupture trace constraints. We used the southwest–northeast-striking nodal plane of the 4 July 2019 M 6.4 event to constrain the initial representation of the southern Little Lake fault (SLLF), both in terms of location and orientation. The eastern Little Lake fault (ELLF) was constrained by the 5 July 2019 M 7.1 hypocenter and nodal planes of M 4 and larger aftershocks aligned with the main trend of the fault. The approach follows a defined workflow that assigns weights to a variety of geometric constraints. These main constraints have a high weight relative to that of individual hypocenters, ensuring that small aftershocks are applied as weaker constraints. The resulting fault planes can be considered averages of the hypocentral locations respecting nodal plane orientations. For the final representation we added detailed, field-mapped rupture traces as strong constraints. The resulting fault representations are generally smooth but nonplanar and dip steeply. The SLLF and ELLF intersect at nearly right angles and cross on another. The ELLF representation is truncated at the Airport Lake fault to the north and the Garlock fault to the south, consistent with the aftershock pattern. The terminations of the SLLF representation are controlled by aftershock distribution. These new 3D fault representations are available as triangulated surface representations, and are being added to a Community Fault Model (CFM; Plesch et al., 2007, 2019; Nicholson et al., 2019) for wider use and to derived products such as a CFM trace map and viewer (Su et al., 2019).

Parametric Surfaced-Based Geological Reservoir Representation: A Computer Graphics Tool for Improved Decision Making in Immersive Environments

<p>Uncertainties are an inherent part of geological interpretation and immersive rendering has the potential to play a key role in gaining better insights. However, most 3D geological models have a limited possibility of manual, fast and smooth modification in order to make better decisions and interpretations. Here we present examples of parametric surface representations which use control points as a possibility to bring interactivity to geological modelling in immersive frameworks.</p><p>In fact, using 2D surfaces of 3D solid objects is a typical representation of 3D models. Two of the major ways for surface representation in computer graphics are implicit representations and parametric surface representations. Parametric surface representations, unlike implicit representations, are based on control points. Manipulating these control points makes it easy and intuitive to modify geological models smoothly and fast, with a potential to more interactive decision-making.</p><p>We present two different examples of parametric surface approaches; Spline Surfaces and Subdivision Surfaces. Spline surfaces, e.g. Bezier or NURBS surfaces, are a popular and common standard for CAD (Computer-Aided Design). Also, these surfaces are on the basis of parametric- based curves and a set of weighted control points. Subdivision Surfaces define smooth surfaces after a series of refinement which can be controlled by control points. Subdivision surfaces are not only a popular method for making free form models but also a common tool in animation, computer games and entertainment industry.</p><p>Recently, research has been done based on using spline surfaces to model diverse geological structures and reservoirs. Similar to applications in computer graphics, using these methods in geological modelling can have specific considerations. Model refinement (e.g. adding new control points) and the requirement of many patches with geometrical constraints for the representation of complex geometries are some of the main difficulties of using spline surfaces. In this presentation, we will discuss several of these aspects and show two promising and controllable techniques for intuitive use of parametric surface-based representations in 3D geological and reservoir modelling.</p>

Slicing heterogeneous solid using octree-based subdivision and trivariate T-splines for additive manufacturing

Purpose Majority of the existing direct slicing methods have generated precise slicing contours from different surface representations, they do not carry any interior information. Whereas, heterogeneous solids are highly preferable for designing and manufacturing sophisticated models. To directly slice heterogeneous solids for additive manufacturing (AM), this study aims to present an algorithm using octree-based subdivision and trivariate T-splines. Design/methodology/approach This paper presents a direct slicing algorithm for heterogeneous solids using T-splines, which can be applied to AM based on the fused deposition modeling (FDM) technology. First, trivariate T-splines are constructed using a harmonic field with the gradient direction aligning with the slicing direction. An octree-based subdivision algorithm is then used to directly generate the sliced layers with heterogeneous materials. For FDM-based AM applications, the heterogeneous materials of each sliced layer are discretized into a finite number of partitions. Finally, boundary contours of each separated partition are extracted and paired according to the rules of CuraEngine to generate the scan path for FDM machines equipped with multi-nozzles. Findings The experimental results demonstrate that the proposed algorithm is effective and reliable, especially for solid objects with multiple materials, which could maintain the model integrity throughout the process from the original representation to the final product in AM. Originality/value Directly slicing heterogeneous solid using trivariate T-splines will be a powerful supplement to current technologies in AM.