A Review of a B-spline based Volumetric Representation: Design, Analysis and Fabrication of Porous and/or Heterogeneous Geometries

The needs of modern (additive) manufacturing (AM) technologies can no longer be satisfied by geometric modeling tools that are based on boundary representations (B-reps) - AM requires the representation and manipulation of interior heterogeneous fields and materials. Further, while the need for a tight coupling between design and analysis has been recognized as crucial almost since geometric modeling (GM) was conceived, contemporary GM systems only offer a loose link between the two, if at all. For more than half a century, the (trimmed) Non-Uniform Rational B-spline (NURBs) surface representation has been the B-rep of choice for virtually all the GM industry. Fundamentally, B-rep GM has evolved little during this period. In this work, we review almost a decade of research and development in extending this boundary representation to a B-spline based, volumetric representation (V-rep) that successfully confronts the existing and anticipated design, analysis, and manufacturing foreseen challenges. We have extended all fundamental B-rep GM operations, such as primitive and surface constructors, and Boolean operations, to trimmed trivariate V-reps. This enables the much-needed tight link between the designed geometry and (iso-geometric) analysis on one hand and the full support of (additive) manufacturing of porous, (graded-) heterogeneous and anisotropic geometries, on the other. Examples and applications of V-rep GM, that span design, analysis and optimization, and AM, of lattice- and micro-structure synthesis as well as graded-heterogeneity, are demonstrated, with emphasis on AM

Heterogeneous Parametric Trivariate Fillets

Blending and filleting are well established operations in solid modeling and computer-aided geometric design. The creation of a transition surface which smoothly connects the boundary surfaces of two (or more) objects has been extensively investigated. In this work, we introduce several algorithms for the construction of, possibly heterogeneous, trivariate fillets, that support smooth filleting operations between pairs of, possibly heterogeneous, input trivariates. Several construction methods are introduced that employ functional composition algorithms as well as introduce a half Volumetric Boolean sum operation. A volumetric fillet, consisting of one or more tensor product trivariate(s), is fitted to the boundary surfaces of the input. The result smoothly blends between the two inputs, both geometrically and material-wise (properties of arbitrary dimension). The application of encoding heterogeneous material information into the constructed fillet is discussed and examples of all proposed algorithms are presented

Synthesis of 3D jigsaw puzzles over freeform 2-manifolds

We present a simple algorithm for synthesizing 3D jigsaw puzzles from arbitrary 3D freeform 2-manifold geometric models represented with trimmed NURBS surfaces. The construction algorithm is based on a few conventional geometric operations on freeform curves and surfaces. In particular, we need to compute the offset of freeform NURBS surfaces (for thickening the 2-manifold surfaces) and the functional composition of a univariate curve representation to a bivariate rational surface (for breaking up a 3D model into curved jigsaw tiles). It is thus almost straightforward to convert the proposed algorithm to a practical system using standard tools available in B-rep based geometric modeling systems, that employ trimmed NURBS surfaces. We demonstrate the effectiveness of the proposed approach by fabricating several test sets of 3D jigsaw puzzles for freeform solids consisting of trimmed NURBS surface models

Euclidean Offset and Bisector Approximations of Curves over Freeform Surfaces

The computation of offset and bisector curves/surfaces has always been considered a challenging problem in geometric modeling and processing. In this work, we investigate a related problem of approximating offsets of curves on surfaces (OCS) and bisectors of curves on surfaces (BCS). While at times the precise geodesic distance over the surface between the curve and its offset might be desired, herein we approximate the Euclidean distance between the two. The Euclidean distance OCS problem is reduced to a set of under-determined non-linear constraints, and solved to yield a univariate approximated offset curve on the surface. For the sake of thoroughness, we also establish a bound on the difference between the Euclidean offset and the geodesic offset on the surface and show that for a C2 surface with bounded curvature, this difference vanishes as the offset distance is diminished. In a similar way, the Euclidean distance BCS problem is also solved to generate an approximated bisector curve on the surface. We complete this work with a set of examples that demonstrates the effectiveness of our approach to the Euclidean offset and bisector operations

Conformal Microstructure Synthesis in Trimmed Trivariate based V-reps

We present a complete microstructure tiling paradigm for V-rep models, or volumetric models consisting of trimmed trivariates. Existing methods [10] are employed to tile individual primitive tensor product trivariates in a conformal way, only to handle the intersection of the microstructure tiles with the trimming surfaces of the trivariates. One-to-one and two-to-one bridging tiles are then constructed along the trimmed zones, while tile-clipping is completely avoided. The Boolean operation cases of subtraction, intersection and union are considered. The result is a set of, regular in the interior, possibly heterogeneous, trivariates, that defines the whole microstructure arrangement. This result is fully compatible with iso-geometric analysis as well as heterogeneous additive manufacturing. Examples are presented, including of 3D printed heterogeneous microstructures

Loading rate sensitivity of open hole composites in compression

The results are reported of an experimental study on the compressive, time-dependent behavior of graphite fiber reinforced polymer composite laminates with open holes. The effect of loading rate on compressive strength was determined for six material systems ranging from brittle epoxies to thermoplastics at both 75 F and 220 F. Specimens were loaded to failure using different loading rates. The slope of the strength versus elapsed time-to-failure curve was used to rank the materials' loading rate sensitivity. All of the materials had greater strength at 75 F than at 220 F. All the materials showed loading rate effects in the form of reduced failure strength for longer elapsed-time-to-failure. Loading rate sensitivity was less at 220 F than the same material at 70 F. However, C12000/ULTEM and IM7/8551-7 were more sensitive to loading rate than the other materials at 220 F. AS4/APC2 laminates with 24, 32, and 48 plies and 1/16 and 1/4 inch diameter holes were tested. The sensitivity to loading rate was less for either increasing number of plies or larger hole size. The failure of the specimens made from brittle resins was accompanied by extensive delaminations while the failure of the roughened systems was predominantly by shear crippling. Fewer delamination failures were observed at the higher temperature

Kernel-based Construction Operators for Boolean Sum and Ruled Geometry

Boolean sum and ruling are two well-known construction operators for both parametric surfaces and trivariates. In many cases, the input freeform curves in IR2 or surfaces in IR3 are complex, and as a result, these construction operators might fail to build the parametric geometry so that it has a positive Jacobian throughout the domain. In this work, we focus on cases in which those constructors fail to build parametric geometries with a positive Jacobian throughout while the freeform input has a kernel point. We show that in the limit, for high enough degree raising or enough refinement, our construction scheme must succeed if a kernel exists. In practice, our experiments, on quadratic, cubic and quartic B´ezier and B-spline curves and surfaces show that for a reasonable degree raising and/or refinement, the vast majority of construction examples are successful

Precise Hausdorff distance computation for freeform surfaces based on computations with osculating toroidal patches

We present an efficient algorithm for computing the precise Hausdorff Distance (HD) between two freeform surfaces. The algorithm is based on a hybrid Bounding Volume Hierarchy (BVH), where osculating toroidal patches (stored in the leaf nodes) provide geometric properties essential for the HD computation in high precision. Intrinsic features from the osculating geometry resolve computational issues in handling the cross-boundary problem for composite surfaces, which leads to the acceleration of HD algorithm with a solution (within machine precision) to the exact HD. The HD computation for general freeform surfaces is discussed, where we focus on the computational issues in handling the local geometry across surface boundaries or around surface corners that appear as the result of gluing multiple patches together in the modeling of generic composite surfaces. We also discuss how to switch from an approximation stage to the final step of computing the precise HD using numerical improvements and confirming the correctness of the HD computation result. The main advantage of our algorithm is in the high precision of HD computation result. As the best cases of the proposed torus-based approach, we also consider the acceleration of HD computation for freeform surfaces of revolution and linear extrusion, where we can support real-time computation even for deformable surfaces. The acceleration is mainly due to a fast biarc approximation to the planar profile curves of the simple surfaces, each generated by rotating or translating a planar curve. We demonstrate the effectiveness of the proposed approach using experimental results

Shape Optimization for Temperature Regulation in Extrusion Dies Using Microstructures

Plastic profile extrusion—a manufacturing process for continuous profiles with fixed cross section—requires a complex and iterative design process to prevent deformations and residual stresses in the final product. The central task is to ensure a uniform material velocity at the outlet. To this end, not only the geometry of the flow decisively influences the quality of the outflow but also the temperature profile along the flow channel wall. It is exactly here that this work contributes by presenting a novel design approach for extrusion dies that will allow for optimal temperature profiles. The core of this approach is the composition of the extrusion die through microstructures. The optimal shape and distribution of these microstructures is determined via shape optimization. The corresponding optimization procedure is the main topic of this article. Special emphasis is placed on the definition of a suitable, low-dimensional shape parameterization. The proposed design-framework is then applied to two numerical test cases with varying complexity

Conformal Parametric Microstructure Synthesis for Boundary Representations

The use of lattices and microstructures in geometric design have been recognized as potentially superior to solid structures due to the potential benefits in improved strength-to-weight ratios, better control over heat exchange and heat transfer, and so on. In this work, we present a construction scheme to create parametric microstructures in a boundary representation (B-rep) model, M, that are conformal to an arbitrary specification, including the boundary of M. Given a B-rep model, M, either a polygonal or trimmed-spline based, a cage, T, is constructed around M to guide the synthesis of the microstructures in M. Micro-elements are synthesized following T, and verified to be inside M while bridging tiles are added as necessary. These parametric micro-elements can be heterogeneous in their material content, as well as locally vary in their geometric properties. We demonstrate these abilities with example microstructures synthesized from both polygonal B-rep models and spline-based B-rep solids, including 3D printed parts