Non-uniform Rational Basis Spline Research Articles

Handling noise and overfitting in surrogate models based on non-uniform rational basis spline entities

This paper proposes a general and systematic approach to generate a surrogate model based on non-uniform rational basis spline (NURBS) hyper-surfaces as a solution of a constrained non-linear programming problem, which is solved through a gradient-based algorithm by considering two formulations. The first formulation, which is the general one, is prone to the curse of dimensionality. The second one separates dimensions in the NURBS domain and allows saving computational time and resources. The optimisation process consists of three steps, which are solved sequentially. Each step is characterised by different design variables and solving strategies. The gradient of the cost function, for both formulations and for each optimisation step, is derived analytically by exploiting the properties of NURBS entities, hence enhancing computational efficiency and accuracy. One of the main features of the presented methodology lies the incorporation of a smoothing term in the problem formulation. This term plays a crucial role in controlling overfitting phenomenon and attenuating local oscillations within the dataset. These oscillations may be due to various sources, such as inherent noise, model instabilities, measurement uncertainties, or numerical precision limitations. The effectiveness of the proposed approach is tested on both analytical benchmark problems and real-world engineering problems taken from the literature.

Trends, Problems, and Solutions from Point Cloud via Non-Uniform Rational Basis Spline to Building Information Modelling: Bibliometric and Systematic Study

Building Information Modelling (BIM) has found applications not only in the planning and designing of new buildings but also in project monitoring, record-keeping, and analyses of existing structures. In addition to its primary application, information modelling spreads to other areas, developing solutions for their specific uses. This study engaged a mixed-method literature review according to the extent of the pool of knowledge for two research steps between three concepts—from point cloud to NURBS and then from NURBS to BIM. In the first step, the keywords point cloud and NURBS were analysed using scientometric methods. In the second step, a systematic analysis of the content of works obtained with the keywords NURBS and BIM is presented, extracting problems and proposed solutions for information transfer technology. The results of a quantitative analysis identified major trends, (1) research is distributed in interdisciplinary and multidisciplinary areas and historical (heritage) modelling is the highlighted one, (2) development of technologies for object surveying, and (3) the application of data in different engineering fields, while a qualitative analysis points at problems in (1) model building, (2) interoperability, and (3) automatization. Solutions such as mixed models, multi-layered models, a mix of formats, or bridging elements (semantics, proto-model) are proposed.

Strength-based topology optimisation of anisotropic continua in a CAD-compatible framework

In this paper, a new method to address strength-based topology optimisation (TO) problems is proposed. Specifically, failure criteria for anisotropic materials are integrated into a TO algorithm making use of Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field describing the topology of the continuum, thus providing solutions that can be easily exported to any computer-aided design software. The notion of failure load factor is introduced to obtain optimised topologies that do not depend on the magnitude of the applied loads. A unified formulation of the main phenomenological criteria for anisotropic materials, i.e., Tsai–Wu, Hoffman and Tsai–Hill criteria, is used and typical issues related to stress-based TO problems, such as local behaviour and singularity of stresses, are handled thanks to: the properties of NURBS blending functions, a special adaptive χ-norm aggregation function that avoids overflow/underflow issues, and a modified pq relaxation approach. The effectiveness of the proposed approach is demonstrated on 2D and 3D problems. A sensitivity analysis of the optimised topologies to the integer parameters involved in the definition of the NURBS entity is performed. Moreover, the influence of the penalisation scheme used for the elasticity and the stress tensors, the failure criterion, and the orientation of the material on the optimal solution is also investigated.

Topology and anisotropy optimisation of continua using non-uniform rational basis spline entities

This paper presents a novel method to address the challenge of concurrent topology and anisotropy optimisation of variable-stiffness anisotropic structures with variable thickness from additive manufacturing technologies. The proposed methodology relies on the multi-level optimisation strategy based on non-uniform rational basis spline (NURBS) entities, density-based method for topology optimisation and the polar formalism to describe the anisotropy of the continuum. Specifically, NURBS entities are used to describe both the density field and the distribution of the polar parameters and the thickness across the design domain. Furthermore, the properties of NURBS entities are efficiently exploited to derive the gradient of the physical responses involved in the problem formulation. The proposed method is applied to the problem of maximising the structural stiffness, while fulfilling design requirements on the lightness and on the feasibility of the polar parameters. In this context, the problem is formulated in the most general case considering inhomogeneous Neumann–Dirichlet boundary conditions. The effectiveness of the approach is tested on benchmark problems taken from literature subjected to both in-plane and out-of-plane loads. Particularly, a large campaign of sensitivity analyses has been carried out to examine the impact of integer parameters of the NURBS entity, the penalty scheme used for the element stiffness matrix, inhomogeneous Neumann–Dirichlet boundary conditions, optimisation strategy (sequential versus concurrent) and initial guess on the optimised solution.

Design of multi-material structures using material jetting technology: Topology optimisation, numerical analysis and experiments

This paper presents a thorough experimental/numerical validation of optimised multi-material structures fabricated by material jetting technology. The proposed methodology uses, on the one hand, non-uniform rational basis spline (NURBS) entities to represent the geometric descriptor associated with each material phase constituting the continuum and, on the other hand, a general multi-phase material interpolation scheme to penalise the stiffness tensor of the structure. Two design requirements are included in the problem formulation: the lightness and the minimum length scale of each material phase. The influence of the integer parameters intervening in the definition of the NURBS entity and the influence of different combinations of material phases on the optimised solutions are investigated. The proposed approach is applied to 2D and 3D benchmark structures subjected to prescribed displacements representative of a three-point bending test. Based on the result of the topology optimisation process one of the optimised solutions, balancing the requirements of structural stiffness, lightness, and manufacturing constraints, is selected, manufactured and tested. A comparison between experimental and numerical results (obtained by non-linear analyses) is carried out to show the effectiveness of the approach.

A topology optimization method for problems with design-dependent loads based on non-uniform rational basis spline entities

In this paper, topology optimization (TO) problems of structures subjected to design-dependent loads are formulated in the context of a special density-based TO approach wherein a Non Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor. Unlike classical density-based TO approaches, the NURBS-density-based method allows representing the pseudo-density field through a purely geometric computer-aided design compatible entity. In this context, the problem is formulated in the most general case of inhomogeneous Neumann-Dirichlet boundary conditions. Moreover, a thorough study of the penalty function of the design-dependent loads is carried out to investigate its effect on the optimized topologies and overcome the singularity effect related to the zones characterized by low values of the pseudo-density field. Finally, a wide campaign of sensitivity analyses is conducted to investigate the influence of the integer parameters of the NURBS entity, of the combination of design-depedent loads and inhomogeneous Neumann-Dirichlet boundary conditions, and of the concentrated load on the optimized topology. The effectiveness of the approach is tested on 2D and 3D benchmark problems.

A NURBS-based IGA using zig-zag plate theory for nonlinear passive/semi-active damping analysis of laminated FG-CNTRC plates

This study introduces an efficient and versatile numerical approach for simulating passive/semi-active damping treatments on laminated structures composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC) plates. This novel approach integrates an isogeometric finite element formulation that leverages basic functions generated from Non-Uniform Rational Basis Spline (NURBS) with the Murakami zig-zag theory, Von Kármán’s nonlinear strain–displacement model, and the electro-mechanical coupling properties of the constraining layers. The proposed isogeometric formulation can be transformed into the Laplace domain using the Golla–Hughes–McTavish model to successfully simulate the time-dependent passive/semi-active damping mechanism of viscoelastic materials. Subsequently, results obtained in the Laplace domain are reconverted into the time domain using inverse techniques. The validation of the proposed model through two numerical examples showcases its high reliability and demonstrates the feasibility of simulating and controlling FG-CNTRC plates with various geometries within a Computer-Aided Design-Computer-Aided Engineering (CAD-CAE) analysis framework.

A multi-physical coupling isogeometric formulation for nonlinear analysis and smart control of laminated CNT-MEE plates

This paper presents a novel and comprehensive numerical approach for nonlinear analysis and smart damping control in laminated functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plate structures, taking into account multiple physical fields. The approach employs a multi-physical coupling isogeometric formulation based on Non-Uniform Rational Basis Spline (NURBS) to analyze the behavior of laminated FG-CNTMEE plate structures. It incorporates a refined zig-zag theory to accurately capture the Von Kármán nonlinear strain–displacement relationship, as well as the magneto-electro-elastic coupling properties of laminate layers. To achieve nonlinear damped responses, the smart constrained layer damping (SCLD) treatment is applied. Furthermore, to enable smart control of structures using the time-dependent passive/semi-active damping mechanism of viscoelastic materials, the formulation is transformed into the Laplace domain using the Golla–Hughes–McTavish model. The results are then converted back to the time domain through inverse techniques. The proposed approach has been rigorously validated through verification work and demonstrated its effectiveness in simulating and controlling FG-CNTMEE plates in two numerical examples.

Spline Search for Slip Surfaces in 3D Slopes

A novel method involving the transformation of spline surfaces is introduced to search for the critical slip surface in a three-dimensional (3D) slope, which corresponds to the minimum limit equilibrium method factor of safety for overall slope stability. A slipping surface in a slope can be represented as the intersection of any continuous geometrical entity with the slope topography, over which the mass of sliding soil is discretized to solve for the factor of safety satisfying given equilibrium conditions. Traditionally, many researchers have searched for a critical ellipsoidal or spherical surface, or surfaces formed using other simple shapes. However, the critical slip surface in complicated cases, for example, a landslide, is seldom purely ellipsoidal or spherical, which leads to overestimations of the true factor of safety in a slope. To provide greater flexibility for transforming the shape of the slip surfaces during a global search, the geometry representing the slip surface is assumed to be in the form of a nonuniform rational basis spline (NURBS) surface in this paper. The proposed method involves varying the parameters of a parametric exponential function, which spawns control points within its domain to create NURBS surfaces. The parameters in the exponential function are varied to transform the slipping surface using a metaheuristic search algorithm, such as particle swarm optimization. A major advantage of the proposed method is that the final spline surface in the search is formulated such that it can then be locally optimized using surface altering optimization methods by adjusting the locations of its control points.

Deep learning aided topology optimization of phononic crystals

In this work, a novel approach for the topology optimization of phononic crystals based on the replacement of the computationally demanding traditional solvers for the calculation of dispersion diagrams with a surrogate deep learning (DL) model is proposed. We show that our trained DL model is ultrafast in the prediction of the dispersion diagrams, and therefore can be efficiently used in the optimization framework.The main novelty of the proposed approach relies on the use of non-uniform rational basis spline (NURBS) curves instead of pixels and/or mesh elements to control the shape of the unit cells of phononic crystals. The surrogate DL model is combined with a genetic algorithm serving as a topology optimization tool. The validity of the approach is shown in the case of phononic crystals made of a continuous matrix with cavities. Several objective functions have been tested as an alternative to the most common gap to mid-gap ratio. This allowed us to obtain interesting phononic crystal geometries which can be easily additively manufactured.The proposed method applies to problems involving inverse design and can open new avenues in the design of computer-assisted periodic structures.

General boundary conditions implementation for the static analysis of anisotropic doubly-curved shells resting on a Winkler foundation

In the present work an Equivalent Single Layer (ESL) formulation is proposed for the static analysis of doubly-curved anisotropic structures of arbitrary geometry and variable stiffness resting on a Winkler elastic foundation. In-plane and out-of-plane general distributions of linear elastic springs are provided for the implementation of general external constraints along the edges of the structure. The structure is geometrically described accounting for the principal curvatures of the shell object of analysis. A generalized set of blending functions based on Non-Uniform Rational Basis Spline (NURBS) curves is adopted so that arbitrary shaped structures can be modelled with the same approach. The fundamental governing equations are obtained in terms of displacement field unknowns, which has been effectively described accounting for a unified formulation based on the minimum potential energy principle. General anisotropic lamination schemes are considered, setting a general orientation of each lamina, as well as all possible material symmetries. The numerical implementation is performed by means of the Generalized Differential Quadrature (GDQ) method, thus allowing a strong formulation of the structural problem. A series of validation examples is performed on shells with zero, single and double curvatures in which the static structural response provided with the proposed formulation has been compared to that obtained from a refined three-dimensional finite element model, showing a great accordance between these different approaches. The research shows that the employment of higher order theories, together with the GDQ method, allows to obtain very accurate results with a reduced computational cost, compared to finite element simulations.

Multipatch isogeometric analysis for geometrically exact shell elements using B-bar method and Bézier extraction

In the present study, a multipatch isogeometric analysis method for geometrically exact shell problems is proposed. Generalized curvilinear coordinates are used for the first-order shear-deformable geometrically exact shell elements. This general tensor-based shell formulation can be directly linked to a computer-aided design system such as nonuniform rational basis spline (NURBS), which is represented by two parameters. Nitsche’s method for patch-to-patch connections in conforming or nonconforming meshes is adopted. The compatibility conditions between NURBS patches in Nitsche’s coupling are weakly imposed using the continuity of displacements, rotations, and stress resultants. The final algebraic equation for the solution fields has a simple symmetric form with previously derived strain–displacement relation matrices and material modulus for membrane, bending, and transverse shear terms. To alleviate the locking phenomenon, the B-bar method, which utilizes the strain projected onto a lower-order field, is used in the isogeometric shell element. Reduced-order B-splines for strain fields are constructed based on tying points, which eased the locking phenomenon compared with the classical B-bar method. To further alleviate additional shell locking by a higher-order regularity, the Bézier extraction method is adopted to reduce the element regularity. The proposed combination of Nitsche’s coupling and B-bar projection demonstrates the superior accuracy and robustness in the multipatch isogeometric shell examples with a high convergence rate.

A Quasi-Optimal Shape Design Method for Electromagnetic Scatterers Based on NURBS Surfaces and Filter-Enhanced GWO

This paper presents a quasi-optimal and efficient method for designing the shape of electromagnetic (EM) scatterers. Optimizing the EM scatterer shape to achieve satisfactory scattering properties is a crucial, yet challenging problem due to the complex geometry involved and the high non-linearity in the optimization. In this paper, the geometric complexity is to be handled by modeling EM scatterer shapes as non-uniform rational basis spline (NURBS) surfaces, which enables an easy and intuitive shape control. Using the surfaces’ control points as optimization variables, the shape optimization problem is to be solved with the heuristic optimization strategy, which can effectively handle non-linearity and the associated local optima. Specifically, the recently developed grey wolf optimizer (GWO) is employed. Although GWO can effectively avoid local optima and achieve quasi-optimal results, it requires repeated calls to time-consuming EM simulations. To solve this efficiency issue, a filter-enhanced version of GWO is proposed to reduce the times of calling EM simulations, and an efficient, adaptive remeshing method is proposed to accelerate the numerical analysis process within each EM simulation call. Altogether, they lead to a quasi-optimal and efficient method for optimizing EM scatterers. Its effectiveness has also been validated by several examples.

A direct approach to achieving efficient free-form shells with embedded geometrical patterns

Shells are widely used in architectural and engineering design due to their appealing geometry and efficient structural form. However, designing free-form shells with specific geometrical patterns remains a challenging issue as both aesthetical and mechanical properties need to be considered. Based on structural topology optimization, this paper develops a direct approach to generating high-performance shells with embedded geometrical patterns. In this approach, graph theory and the nonuniform rational basis spline (NURBS) method are combined to produce parametric designs based on polygonal meshes. This approach is capable of optimizing shells with complicated geometry and different thicknesses. The effectiveness of the approach is demonstrated by three representative examples. The techniques developed in this study can be used to design efficient free-form shells for construction.

On multi-material topology optimisation problems under inhomogeneous Neumann–Dirichlet boundary conditions

This work deals with the topology optimisation of structures made of multiple material phases. The proposed approach is based on non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the continuum, and an improved multiphase material interpolation (MMI) scheme to penalise the stiffness tensor of the structure. In this context, the problem is formulated in the most general case by considering inhomogeneous Neumann–Dirichlet boundary conditions and by highlighting the differences between two different problem formulations. The first one uses the work of applied forces and displacements as cost function and the resulting optimisation problem is not self-adjoint. The second one considers the generalised compliance (related to the total potential energy), and the resulting optimisation problem is self-adjoint. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the boundary of the topological descriptors of the material phases composing the structure. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, an extensive campaign of numerical analyses is conducted to investigate the influence of the type of geometric descriptor, of the integer parameters involved in the definition of the NURBS entity, of the type of cost function, of the type of lightness requirement, of the number and type of material phases, of the applied boundary conditions on the optimised topology. • Topology optimisation of multi-material media under mixed boundary conditions • The generalised compliance is used as a measure of the structural stiffness • The work of external actions is not a self-adjoint function (unlike the compliance) • The geometrical descriptor of each material phase is a NURBS hyper-surface • Influence of NURBS parameters and number of materials on the optimal solution

Topology optimisation of architected cellular materials from additive manufacturing: Analysis, design, and experiments

This work deals with an experimental/numerical validation of the optimised topologies found through a special density-based topology optimisation (TO) method wherein the topological descriptor, i.e., the pseudo-density field, is represented through a non-uniform rational basis spline (NURBS) hyper-surface. The framework is that of multi-scale TO methods to design architected cellular materials (ACMs). Specifically, in the most general case, the topological variables are defined at the scale of the representative volume element (RVE) of the ACM and at the macroscopic scale of the structure. The transition among scales is performed via a numerical homogenisation scheme based on the strain energy of elements. The proposed formulation exploits the properties of NURBS entities to determine the relationships occurring among the topological variables defined at different scales to correctly state the optimisation problem and to satisfy the hypotheses at the basis of the homogenisation method. Three design cases are considered: in the first one, TO is performed only at the macroscopic scale; in the second one, TO is performed only at the RVE scale; in the last one, TO is performed simultaneously at both scales. Multiple design requirements related to lightness, scale separation condition (to ensure the validity of the results of the homogenisation method) and minimum printable size are included in the problem formulation. Particularly, the last two requirements are implicitly satisfied by controlling the integer parameters of the NURBS entity (describing the pseudo-density field at each scale) without introducing explicit optimisation constraints. The multi-scale TO strategy is applied to a structure made of ACM subject to three-point bending test-like boundary conditions: for each design case, the optimised topology is manufactured through stereo-lithography and a comparison between experimental and numerical results (obtained through non-linear analysis conducted a posteriori on the optimised topology) is performed to assess the effectiveness of the approach.

Dynamic analysis of anisotropic doubly-curved shells with general boundary conditions, variable thickness and arbitrary shape

In the present contribution a general formulation is proposed to account for general boundary conditions within the dynamic analysis of anisotropic laminated doubly-curved shell having arbitrary shape and variable thickness. Different analytical expressions are considered for the shell thickness variation along the geometrical principal directions, and the distortion of the physical domain is described by a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) curves. Mode frequencies and shapes are determined employing higher-order theories within an Equivalent Single Layer (ESL) framework. The related fundamental relations are tackled numerically by means of the Generalized Differential Quadrature (GDQ) method. The dynamic problem is derived from the Hamiltonian Principle, leading to a strong formulation of the governing equations. General external constraints are enforced along the edges of the shell employing a distribution of linear springs distributed on the faces of the three-dimensional solid and accounting for a spatial coordinate-dependent stiffness along both in-plane and out-of-plane directions. Moreover, a Winkler-type foundation with general distribution of linear springs is modelled on the top and bottom surfaces of the shell. A systematic set of numerical examples is carried out for the validation of the proposed theory by comparing mode frequencies with predictions from refined three-dimensional finite element analyses. Finally, we perform a sensitivity analysis of the dynamic response of mapped curved structures for different spring stiffnesses and general external constraints, according to various kinematic assumptions.

High-fidelity standard model reconstruction and verification of an airliner based on point clouds.

Computational fluid dynamics (CFD) numerical simulation as the primary research tool is particularly essential for its credibility during the aerodynamic design of aircraft. To further promote CFD verification and validation on the airliner, a high-fidelity model reconstruction of the airliner is fundamental. Based on this, we put forward a novel framework, to our best knowledge, to reconstruct a high-fidelity standard model for an airliner efficiently, and the feasibility and accuracy of these reconstructed models are accessed by the CFD simulation-based validation method. First and foremost, a laser scanner was placed at each station around the airliner to scan and acquire multiview point clouds. Afterwards, the truncated least-squares-based algorithm was adopted to register these point clouds entirely. Additionally, we fitted the nonuniform rational basis spline surface based on the least-squares progressive and iterative approximation algorithm. Finally, CFD simulation results were compared and analyzed with the aerodynamic data obtained by the aircraft manufacturer under the same Mach number of the uniform model. It turns out that the coincidence between them is high, and the changing trend is basically consistent. Hence, this method is highly feasible and can establish a high-fidelity standard model of an airliner with unified high and low speeds so that its appearance, test data, and research results can be adopted as the standard data for CFD verification and validation.

Multi-scale design of multi-material lattice structures through a CAD-compatible topology optimisation algorithm

This work deals with the multi-scale topology optimisation (TO) of multi-material lattice structures. The proposed approach is based on: non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the representative volume element (RVE), an improved multiphase material interpolation (MMI) scheme to penalise the element stiffness tensor of the multi-material RVE, the strain energy-based homogenisation method (SEHM) to carry out the scale transition. In this context, the design requirements are defined at different scales and their gradient is evaluated by exploiting the properties of the NURBS entities and of the SEHM. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the topological descriptors of the material phases composing the RVE. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, a sensitivity analysis of the optimised configuration of the RVE to the parameters tuning the shape of the NURBS entity is conducted. Finally, the influence of the starting point and of the macroscopic loads on the optimal solution is investigated.

Hands-free printed door opener to limit the spread of Coronavirus: Design through topology optimization

Recently, the necessity of reducing the probability of spread of viruses has fostered the creativity of engineers to develop tools that would allow actions of every-day life to be executed differently. Moreover, the maturity of the Fused Filament Fabrication (FFF) technology and the associated low costs has allowed creative solutions to be produced and used in real-life applications. A distinctive example is represented by the common action of opening a door. Since hands are a typical vector of contamination for viruses such as Coronavirus, hands-free devices aim at making use of the existing structure and kinematic to complete the same action in a different fashion. Typically, the mechanical and manufacturing requirements of these devices include a suitable stiffness-to-mass ratio, a reduced printing time as well as the minimization of supports which need to be removed in a post-printing phase. To tackle all these requirements a dedicated topology optimization (TO) method can be used since the preliminary design phase. Several design requirements of different nature can be included in the problem formulation: mechanical ones, like mass and stiffness, and manufacturing ones, like drawing direction or minimum member size. In this paper, a feasibility study on a hands-free 3D printed door opener has been carried out by means of the Solid Isotropic Material with Penalization (SIMP) method for TO and its CAD-compatible variant, i.e., the SIMP approach reformulated in the framework of non-uniform rational basis spline hyper-surfaces. The aim of the study is to identify optimal solutions to be adapted to a real-case scenario wherein different loading cases and manufacturing constraints are evaluated. Different optimal solutions are obtained, reconstructed to be compatible with CAD environment and the optimized geometry numerically assessed. Finally, the optimal solutions are also evaluated with respect to indicators such as printing time, total filament mass and mass of the supports required by the printing process.