Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures

Numerical Meth Engineering

2019

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Summary The need to compute the intersections between a line and a high‐order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a noniterative method for computing intersections by solving a matrix singular value decomposition and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one‐shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton‐Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree q of the shape functions and is 2q × 3q for curves and 6q2 × 8q2 for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high‐order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/.

{bold-italicA}^{□}

and

{bold-italicB}^{□}

of the largest size, eg, the first four columns of A and B , can be considered

{bold-italicϕ}^{□} ((), {bold-italicA}^{□} - ξ {bold-italicB}^{□}) = bold0 .

Although there can only be three intersection points for a cubic curve, this problem has four eigenvalues and eigenvectors.…”

Section: Intersection Of Lines With Curvesmentioning

confidence: 99%

Section: Illustrative Examplementioning

confidence: 99%

A noniterative method for robustly computing the intersections between a line and a curve or surface

Xiao

Busé

Numerical Meth Engineering

2019

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“…44 To close this gap, we propose a shape and topology optimization approach by combining topology optimization of the lattice with shape optimization of the entire structure. Building on our earlier work on isogeometric design and analysis of lattice-skin structures, 45 the lattice is modeled as a pin-jointed truss and the skin as a Kirchhoff-Love shell. The lattice consists of a large number of cells, which in turn consist of a small number of struts connected by pins that do not transfer moments.…”

Section: Introductionmentioning

confidence: 99%

Infill topology and shape optimization of lattice‐skin structures

Xiao

Numerical Meth Engineering

2021

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Lattice-skin structures composed of a thin-shell skin and a lattice infill are widespread in nature and large-scale engineering due to their efficiency and exceptional mechanical properties. Recent advances in additive manufacturing, or 3D printing, make it possible to create lattice-skin structures of almost any size with arbitrary shape and geometric complexity. We propose a novel gradient-based approach to optimizing both the shape and infill of lattice-skin structures to improve their efficiency further. The respective gradients are computed by fully considering the lattice-skin coupling while the lattice topology and shape optimization problems are solved in a sequential manner. The shell is modeled as a Kirchhoff-Love shell and analyzed using isogeometric subdivision surfaces, whereas the lattice is modeled as a pin-jointed truss. The lattice consists of many cells, possibly of different sizes, with each containing a small number of struts. We propose a penalization approach akin to the SIMP (solid isotropic material with penalization) method for topology optimization of the lattice. Furthermore, a corresponding sensitivity filter and a lattice extraction technique are introduced to ensure the stability of the optimization process and to eliminate scattered struts of small cross-sectional areas. The developed topology optimization technique is suitable for nonperiodic, nonuniform lattices. For shape optimization of both the shell and the lattice, the geometry of the lattice-skin structure is parameterized using the free-form deformation technique. The topology and shape optimization problems are solved in an iterative, sequential manner. The effectiveness of the proposed approach and the influence of different algorithmic parameters are demonstrated with several numerical examples.

“…Trimming involves the computation of the intersection between spline surfaces with other surfaces or curves. The respective nonlinear root-finding problems look deceptively simple but are extremely hard to robustly solve and lead to non-watertight geometries [1][2][3]. In the analysis context, the non-watertight geometries obtained from trimming pose unique challenges.…”

Section: Introductionmentioning

confidence: 99%

Manifold-based isogeometric analysis basis functions with prescribed sharp features

Zhang

Computer Methods in Applied Mechanics and Engineering

2020

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We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. This makes them particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks, which require C 1 continuous basis functions that are C 0 continuous across the kinks. We demonstrate the convergence and utility of the new basis functions with linear and nonlinear beam, plate and shell examples.