Trimming Curves Research Articles

Untrimming: Precise conversion of trimmed-surfaces to tensor-product surfaces

Trimmed B-spline surfaces are common in the geometric computer aided design (CAD) community due to their capability to represent complex shapes that can not be modeled with ease using tensor product B-spline and NURBs surfaces. However, in many cases, handling trimmed-surfaces is far more complex than tensor-product (non-trimmed) surfaces. Many algorithms that operate on tensor-product surfaces, such as algorithms toward rendering, analysis and manufacturing, need to be specially adapted to consider the trimming domains. Frequently, these special adaptations result in lack of accuracy and elevated complexity. In this paper, we present an algorithm for converting general trimmed surfaces into a set of tensor-product (typically B-spline) surfaces. We focus on two algorithms to divide the parametric space of the trimmed surface into four-sided quadrilaterals with freeform curved boundaries, which is the first step of the algorithm. Then, the quadrilaterals are parameterized as planar parametric patches, only to be lifted to the Euclidean space using a surface-surface composition, resulting in tensor product surfaces that precisely tile the input trimmed surface in Euclidean space. The algorithm is robust and precise. We show that we can handle complex, industrial level, objects, with numerous high orders and rational surfaces and trimming curves. Finally, the algorithm provides user control on some properties of the generated tensor-product surfaces.

Cut finite element methods for elliptic problems on multipatch parametric surfaces

We develop a finite element method for the Laplace–Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace–Beltrami operator using a cut finite element method that utilizes Nitsche’s method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L2 norm and also present several numerical examples confirming our theoretical results.

A parameter-free variational coupling approach for trimmed isogeometric thin shells

The non-symmetric variant of Nitsche's method was recently applied successfully for variationally enforcing boundary and interface conditions in non-boundary-fitted discretizations. In contrast to its symmetric variant, it does not require stabilization terms and therefore does not depend on the appropriate estimation of stabilization parameters. In this paper, we further consolidate the non-symmetric Nitsche approach by establishing its application in isogeometric thin shell analysis, where variational coupling techniques are of particular interest for enforcing interface conditions along trimming curves. To this end, we extend its variational formulation within Kirchhoff---Love shell theory, combine it with the finite cell method, and apply the resulting framework to a range of representative shell problems based on trimmed NURBS surfaces. We demonstrate that the non-symmetric variant applied in this context is stable and can lead to the same accuracy in terms of displacements and stresses as its symmetric counterpart. Based on our numerical evidence, the non-symmetric Nitsche method is a viable parameter-free alternative to the symmetric variant in elastostatic shell analysis.

B++ splines with applications to isogeometric analysis

A novel spline, named B++ Splines (Boundary Plus Plus Splines), is developed to address the expression of a trimmed NURBS patch in an analytic form, which is a central idea of surface representation. The presented method converts each trimmed NURBS patch into a B++ spline patch that incorporates of specific boundary points as the boundary presentation. Emphasis is placed on the construction of a new analytic formula of a trimmed NURBS patch defined by the boundary points at the trimming curves and a group of enriched control points. B++ spline basis functions are linearly independent, build a partition of unity and satisfy the Kronecker delta property. Each B++ spline basis function is a linear combination of the basis functions of the trimmed NURBS patch. These properties allow imposing the Dirichlet boundary conditions strongly at the boundary of the trimmed patch without the necessity of modifying the basis functions of the trimmed patch. Isogeometric analysis using B++ splines for two-dimensional elastic solids is also proposed. Several numerical examples are used to demonstrate the reliability of the presented method. The numerical example for the patch test illustrates that the B++ spline patch passes the standard patch test.

Isogeometric topology optimization of shell structures using trimmed NURBS surfaces

In the present research, the isogeometric topology optimization of shell structures is proposed. There have been lots of successful studies on the shape optimization using isogeometric analysis (IGA). However, it is not straightforward to apply the conventional IGA to topology optimization. Topological changes of a domain are hard to be represented due to the tensor-product form of a Non-Uniform Rational B-Spline (NURBS) surface. Since only quadrilateral domains can be handled in the conventional IGA, a topologically complex domain which frequently appears during the optimization process should be built by introducing multiple untrimmed NURBS patches for analysis. Trimmed surface analysis (TSA) is the IGA where trimming techniques are employed, and it can handle a topologically complex domain with a single NURBS patch effectively. In the present work, the basic concept of the two-dimensional TSA is appropriately adapted to shell structures in order to handle the complex topologies. The whole optimization process includes the topological change step and the shape optimization step. The criteria based on the topological derivatives are used for the judgement of new hole creation and the decision of the hole position. Holes are represented by introducing NURBS trimming curves. The trimming curve control points as well as the surface control points are set as design variables for the shape optimization. In the optimization process, updating of the trimming curves is performed in adaptive manners to maintain smooth boundary representation and robust convergence. With numerical examples, it is shown that the proposed method gives appropriate and acceptable shell structure designs. By the inner or outer boundaries of a domain which are described by trimming curves, the smooth material layout is derived without gray scales or checkerboard patterns. Since the same IGES standard is used throughout the process, the final design derived by the present method can be directly communicated with CAD systems without any additional treatments.

Direct immersogeometric fluid flow analysis using B-rep CAD models

We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep.

Weighted T-splines with application in reparameterizing trimmed NURBS surfaces

Abstract To facilitate isogeometric analysis, this paper presents a new type of T-spline named the weighted T-spline, which introduces a new weighting idea to T-spline basis functions. Weighted T-spline basis functions satisfy partition of unity and are linearly independent. In addition, we apply the bicubic weighted T-splines to reparameterize trimmed NURBS surface patches. Edge interval extension is performed to reconstruct the trimming curve on the T-spline surface, and the trimming curve can be exactly preserved. Comparisons with standard T-splines show that the weighted T-splines can decrease the required number of control points and T-mesh elements. The surface error introduced by weighted T-spline basis functions is bounded (within 0.5%), and the error introduced by the trimming curve is constrained within its three-ring neighboring elements. Weighted T-spline models are also applied to solve the linear elasticity problems and Poisson’s equation, demonstrating that they can be used in isogeometric analysis.

Isogeometric analysis of topologically complex shell structures

In the conventional isogeometric analysis, a topologically complex shell structure is required to be modeled using multiple NURBS patches. In the CAD industry, however, the topologically complex shell structure is conveniently and efficiently created via trimming techniques rather than constructing multiple untrimmed NURBS patches. With this feature, the isogeometric analysis which enables to handle the topologically complex shell structure with a single NURBS patch is presented in this paper. In the present method, the information of the topologically complex shell structure composed of the untrimmed surface and trimming curves is directly utilized into isogeometric shell analysis. For numerical integration, a special integration scheme is adopted considering the inside or the outside of the trimmed boundaries which are described by the trimming curves. For the shell formulation, the degenerated shell element based on the Reissner–Mindlin theory is employed. The exact surface normal vectors and their analytic derivatives are adopted into the formulation. The shell formulation is validated with linear elastic benchmark problems. Then linear elastic problems of topologically complex shell structures are dealt with using the proposed procedures, and the effectiveness is illustrated.

Correct resolution rendering of trimmed spline surfaces

Current strategies for real-time rendering of trimmed spline surfaces re-approximate the data, pre-process extensively or introduce visual artifacts. This paper presents a new approach to rendering trimmed spline surfaces that guarantees visual accuracy efficiently, even under interactive adjustment of trim curves and spline surfaces. The technique achieves robustness and speed by discretizing at a near-minimal correct resolution based on a tight, low-cost estimate of adaptive domain griding. The algorithm is highly parallel, with each trim curve writing itself into a slim lookup table. Each surface fragment then makes its trim decision robustly by comparing its parameters against the sorted table entries. Adding the table-and-test to the rendering pass of a modern graphics pipeline achieves anti-aliased sub-pixel accuracy at high render-speed, while using little additional memory and fragment shader effort, even during interactive trim manipulation.

Conversion of trimmed NURBS surfaces to Catmull–Clark subdivision surfaces

This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity.

Local and global analysis of parametric solid sweeps

In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a “local analysis”, which covers parametrizations and singularities, as well as a “global theory” which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the “funnel” which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface–surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and non-decomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.

Spline-based meshfree method with extended basis

In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis.

A contribution to the development of a full flight envelope quasi-nonlinear helicopter simulation

Helicopter modeling is a complex task as helicopters consist of many subsystems (nacelle, rotor, engine, $$\ldots)$$ that are described by coupled differential equations. Controller design requires a model that is accurate over a certain frequency range. Usually, system identification is used to derive the models from flight test data and the model complexity depends on the frequency range of interest. As helicopters have eigenvalues that vary depending on airspeed, system identification is performed at different operating points. Each identified linear model is then valid only in a small region around the corresponding operating point. To arrive at a model that covers the whole operational envelope of the helicopter, the individual models are stitched together. This paper describes the derivation of a quasi-nonlinear model accounting for known nonlinear terms, such as gravity, inertia or trim curves. In addition, the coefficients of the identified linear matrices are interpolated depending on airspeed. Finally, the quasi-nonlinear as well as the linear model are compared against flight test data using model validation techniques. As the quasi-nonlinear model of the Flying Helicopter Simulator is validated, it can be used for a sophisticated controller design.

Filling trim cracks on GPU-rendered solid models

We present an algorithm for improving the rendering appearance of CAD models with trimmed freeform surfaces when evaluated on graphics processing units (GPUs). Rendering on client GPUs allows mechanical CAD to embrace cloud computing by storing a single auto-synchronized model file in the cloud and transferring only minimal data (control points, trim curves, etc.) to the client nodes for local evaluation/rendering. However, current parallel algorithms that directly evaluate and render trimmed surfaces by masking the trims, without tessellating along the trim curves, suffer from “cracks” along the trim boundaries. We have developed a hybrid CPU–GPU algorithm to remove these artifacts in the rendering stage for a smooth, color- and shading-matched appearance. After dynamically detecting the cracks, our algorithm selectively fills in the affected pixels using a GPU fragment program, while avoiding artifacts at silhouettes. We have implemented this algorithm to demonstrate improvements in the appearance of solid models directly evaluated and rendered on the GPU.

A robust efficient tracing scheme for triangulating trimmed parametric surfaces

An efficient, robust parametric trimmed surface triangulation method is presented. Efficiency is gained during trimmed curve tracing by minimising the number of cells processed. Key feature is the efficient tracing algorithm, and knowledge of orientation of the trimming curves is not required. The method is applicable to NURBS surfaces and operates on the untrimmed surface, constructing a rectangular parametric grid onto which the trimming curves are traced. This approach also minimises the occurrence of degenerate triangles and copes with holes independently of the grid size.

Isogeometric topology optimization using trimmed spline surfaces

In the present work, a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed. In the proposed approach, the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by CAD systems is employed for structural response analysis and sensitivity calculation in the topology optimization. The outer and inner boundaries of design models are represented by a spline surface and trimming curves. Design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves. New sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified. The creation of new inner fronts during optimization is allowed for the topological flexibility. An inner front merging algorithm is also presented. The proposed spline based topology optimization is used to solve some benchmarking problems. Design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization. Design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one. It is also shown that post-processing effort for converting to CAD model is eliminated by using the same spline information in numerical analysis and design optimization.

Isogeometric analysis with trimming technique for problems of arbitrary complex topology

Trimming technique is a powerful and efficacious way of endowing an arbitrary complex topology to CAD files created by using NURBS. In the present work, it is shown that any complex multiply-connected NURBS domain can be described by using trimming curves only. Isogeometric analysis for linear elasticity problems of complex topology described in this way is presented. For fully communicative interaction between CAD and CAE, a specific searching algorithm and an integration scheme of trimmed elements are introduced to utilize the IGES files exported from CAD system for Isogeometric analysis. Schemes for imposing essential and traction boundary conditions on trimming curves are presented. It has been demonstrated that with the presented schemes trimmed cases in any complicated situations can be successfully treated. With the examples of complex topology that could be described by employing trimming curves only, effectiveness and robustness of present method are demonstrated.

Direct trimming of NURBS surfaces on the GPU

This paper presents a highly efficient direct trimming technique for NURBS surfaces, which is applicable to tessellation-based rendering as well as ray tracing systems. The central idea is to split the trim curves into monotonic segments with respect to the two parameter dimensions of the surface patches. We use an optimized bisection method to classify a point with respect to each monotonic trim curve segment without performing an actual intersection test. Our hierarchical acceleration structure allows the use of a large number of such curve segments and performs the bisection method only for points contained in the bounding boxes of the curve segments. We have integrated our novel point classification scheme into a GPU-based NURBS ray casting system and implemented the entire trimmed NURBS rendering algorithm in a single OpenGL GLSL shader. The shader can handle surfaces and trim curves of arbitrary degrees, which allows the use of original CAD data without incorporating any approximations. Performance data confirms that our trimming approach can deal with hundreds of thousands of trim curves at interactive rates. Our point classification scheme can be applied to other application domains dealing with complex curved regions including flood fills, font rendering and vector graphics mapped on arbitrary surfaces.

Isogeometric analysis for trimmed CAD surfaces

In this work, an Isogeometric analysis for trimmed CAD surfaces in 2D linear elasticity problem is presented. The main benefit of the proposed method is that no additional modeling for the analysis of the trimmed surface is necessary. As a pioneering attempt to deal with the trimmed surfaces in Isogeometric analysis, the information on the trimming curves and trimmed surfaces exported from CAD system is directly utilized for analysis. For this, trimmed elements are searched through using a projection scheme. For the integration of trimmed elements, NURBS-enhanced integration scheme employed in NEFEM is adopted. For the Isogeometric analysis, the construction of the stiffness matrix based on the spline basis function is presented. In the formulation, the information on the trimming curves is used not only for obtaining integration points but also for calculating the Jacobian. Through error analyses of the verification examples, the robustness and effectiveness of the proposed method are investigated. It is observed that the proposed method gives the theoretical convergence rate. Multiple-trimming-curve problems which are difficult to analyze with conventional Isogeometric analysis are easily treated with the proposed method. An illuminating example is given. In the same example, T-spline local refinement is also employed.

트림 NURBS 곡면의 T-스플라인 유한요소해석

In this present work, spline FEA for the trimmed NURBS surface of the 2D linear elasticity problem is presented. The main benefit of the proposed method is that no additional efforts for modeling of trimmed NURBS surfaces are needed and the information of the trimming curves and trimmed surfaces exported from the CAD system can be directly used for analysis. For this, trimmed elements are searched by using NURBS projection scheme. The integration of the trimmed elements is performed by using the NURBS-enhanced integration scheme. The formulation of constructing stiffness matrix of trimmed elements is presented. In this formulation, the information of the trimming curve is used for calculating the Jacobian as well as for obtaining integration points. The robustness and effectiveness of the proposed method are investigated through various numerical examples.