Search Results (73)

We investigate the application of soft-cell tessellations—a recently discovered class of curved-boundary space-filling shapes—to finite element mesh generation. Using Gmsh and scikit-fem, we compare the solution accuracy for Poisson equation benchmarks on curved domains. The results demonstrate that soft-cell meshes achieve optimal $O\left(h^2\right)$ convergence rates in $L^2$, matching conventional elements. More significantly, we identify a fundamental limitation: coarse polygon boundaries introduce systematic boundary condition (BC) error (∼3%) that does not decrease with mesh refinement. We prove analytically that the BC error scales as $O(1/n^2)$ for n-point polygon boundaries, explaining why doubling boundary points reduces the error by $4\times$. Fine spline boundaries reduce this error by 96%, with the interior solution error reduced by 97.5%. For complex organic shapes, the improvement reaches 56–80%. We establish a connection between the soft-cell softness measure $\sigma$ and FEM accuracy: a higher softness yields a lower BC error. Comparison with Isogeometric Analysis reveals that while IGA achieves exact geometry (${10}^{-16}$ error), fine spline FEM boundaries reduce the geometric error by 5–6 orders of magnitude versus coarse polygons. These results establish that the boundary representation quality fundamentally limits the FEM accuracy on curved domains, making soft-cell representations particularly valuable. Full article

Large temperature gradients encountered in aerospace, energy, and microelectronics systems impose stringent requirements on material thermal performance. Functionally graded materials (FGMs), characterized by a continuous variation in composition and properties, offer significant advantages in regulating heat transfer and mitigating thermal stresses. This review provides a systematic summary of recent progress in heat transfer design optimization of FGMs under large temperature gradient conditions. From a methodological perspective, advancements in structural and compositional optimization, topology optimization, and multi-objective optimization are reviewed. Numerical simulation techniques, including conventional finite element and finite volume methods, as well as emerging approaches such as peridynamics, isogeometric analysis, and meshfree methods, are discussed with an emphasis on multiphysics coupling. In addition, representative applications of FGMs in electronic thermal management, aerospace thermal protection, energy systems, and building energy conservation are reviewed. Current challenges, including idealized modeling assumptions, limited coordination among multiple optimization objectives, and insufficient reliability evaluation in complex service environments, are identified. Finally, future research directions are outlined, highlighting intelligent design methods, multiscale modeling, advanced manufacturing technologies, and multifunctional integration. This review seeks to provide a comprehensive reference for both fundamental research and engineering applications of heat transfer optimization in functionally graded materials. Full article

Orthotropic materials are increasingly employed in advanced thermal systems due to their direction-dependent heat transfer characteristics. Accurate numerical modeling of heat conduction in such media remains challenging, particularly for 3D geometries with nonlinear boundary conditions and internal heat generation. In this study, conventional boundary element method (BEM) and isogeometric boundary element method (IGABEM) formulations are developed and compared for steady-state orthotropic heat conduction problems. A coordinate transformation is adopted to map the anisotropic governing equation onto an equivalent isotropic form, enabling the use of classical Laplace fundamental solutions. Volumetric heat generation is incorporated via the radial integration method (RIM), preserving the boundary-only discretization, while nonlinear Robin boundary conditions are treated using variable condensation and a Newton–Raphson iterative scheme. The performance of both methods is evaluated using a hollow ellipsoidal benchmark problem with available analytical solutions. The results demonstrate that IGABEM provides higher accuracy and smoother convergence than conventional BEM, particularly for higher-order discretizations, which is owing to its exact geometric representation and higher continuity. Although IGABEM involves additional computational overhead due to NURBS evaluations, both methods exhibit similar quadratic scaling with respect to the degrees of freedom. Full article

Isogeometric analysis (IGA) is a numerical methodology for solving differential equations by employing basis functions that preserve the exact geometry of the domain. This approach is based on a class of mathematical functions known as NURBS (Non-Uniform Rational B-Splines). Representing a domain with NURBS entities often requires multiple patches, especially for complex geometries. Bivariate NURBS, defined as tensor products, enforce global refinements within a patch and, in multi-patch models, these refinements are propagated to other model patches. The use of T-Splines with extraordinary points offers a solution to this issue by enabling local refinements through unstructured meshes. The analysis of T-Spline models is performed using a Bézier extraction technique that relies on extraction operators that map Bézier functions to T-Spline functions. When generating a T-Spline model, careful attention is required to ensure that all T-Spline functions are linearly independent—a necessary condition for IGA—in order to form T-Splines that are suitable for analysis. In this sense, this work proposes a methodology to automate the generation of bidimensional unstructured meshes for IGA through T-Splines with extraordinary points. An algorithm for generating unstructured finite element meshes, based on domain decomposition of quadrilateral patches, is adapted to construct T-Spline models. Validation models demonstrate the methodology’s flexibility in generating locally refined isogeometric models. Full article

This paper presents a unified framework for constructing partially unstructured B-spline transfinite finite elements with arbitrary nodal distributions. Three novel, distinct classes of elements are investigated and compared with older single Coons-patch elements. The first consists of classical transfinite elements reformulated using B-spline basis functions. The second includes elements defined by arbitrary control point networks arranged in parallel layers along one direction. The third features arbitrarily placed boundary nodes combined with a tensor-product structure in the interior. For all three classes, novel macro-element formulations are introduced, enabling flexible and customizable nodal configurations while preserving the partition of unity property. The key innovation lies in reinterpreting the generalized coefficients as discrete samples of an underlying continuous univariate function, which is independently approximated at each station in the transfinite element. This perspective generalizes the classical transfinite interpolation by allowing both the blending functions and the univariate trial functions to be defined using non-cardinal bases such as Bernstein polynomials or B-splines, offering enhanced adaptability for complex geometries and nonuniform node layouts. Full article

Nanostructures, such as carbon nanotubes (CNTs), graphene, nanoplates, etc., show behaviors that classical continuum theories cannot capture. At the nanoscale, size effects, surface stresses, and nonlocal interactions become important, so new models are needed to study nanostructures. The main nanomechanics theories that are used in recently published papers include nonlocal elasticity theory (NET), couple stress theory (CST), and nonlocal strain gradient theories (NSGTs). To solve these models, methods such as finite elements, isogeometric analysis, mesh-free approaches, molecular dynamics (MD), etc., are used. Also, this review categorizes and summarizes the major theories and numerical methods used in nanomechanics for the analysis of nanostructures in recently published papers. Recently, machine learning methods have enabled faster and more accurate prediction of nanoscale behaviors, offering efficient alternatives to traditional methods. Studying these theories, numerical models and data driven approaches provide an important foundation for future research and the design of next generation nanomaterials and devices. Full article

While the established theory of computer-aided geometric design (CAGD) suggests that rational Bernstein–Bézier polynomials associated with control points can be used to accurately represent conics and quadrics, this paper shows that the same goal can be achieved in a different manner. More specifically, rational Lagrange polynomials of the same degree, associated with nodal points lying on the true curve or surface, can be combined with appropriate weights to yield equivalent numerical results within a Bézier patch. The specific application of this equivalence to derive weights for Lagrange nodes on conics and quadrics is shown in this paper. Although this replacement may not be crucial for CAGD purposes, it proves useful for the direct implementation of boundary conditions in isogeometric analysis, since it allows the use of nodal values on the exact boundary. Full article

This study proposes an adaptive refinement method based on feature recognition to rapidly obtain solutions of the Reynolds equation. Leveraging an isogeometric analysis (IGA) solution framework supporting local refinement, three natural refinement features tailored to solving the Reynolds equation in fluid lubrication, including two physical features, a pressure value, a pressure gradient, and an element size feature for discretization, are introduced first to identify mesh elements. Then a neural network model is trained on feature data to predict element classifications effectively. Finally, this model is integrated into the adaptive refinement solution framework and validated through simulations. Comparative validation was conducted on two distinct Reynolds equation instances, with the results demonstrating that the proposed algorithm can effectively evaluate refinement regions globally, avoiding issues such as mesh non-conformity often caused by conventional independent element marking algorithms. The distribution of degrees of freedom is more rational, and the parallel prediction model enhances the speed of the refinement solution. Full article

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

We study overlapping additive Schwarz (OAS) preconditioners for the solution of elliptic boundary value problems discretized using isogeometric collocation methods based on generalized B-splines (GB-splines). Through a series of numerical experiments, we demonstrate the scalability of the proposed preconditioning strategy with respect to the number of subdomains, as well as its robustness with respect to the parameters of the isogeometric discretization. Full article

LGP cylinders are necessary for fuel storage and home heating. To avoid material and human risk, it is essential to maintain their structural integrity. Extensive mechanical research studies and physical tests are necessary for its design. This paper investigates the mechanical performance of the storage capacity of an LPG cylinder under static loading. The authors integrate and adapt IGA with the T-Splines function for geometry modeling and numerical analysis in the context of linear elasticity. The main focus is on the strains and stress numerical results. The obtained results are examined and verified with the FEM in Abaqus/Standard. The results found show that the storage capacity of a single cylinder is equivalent to 15 empty cylinders. This study also demonstrates that the T-Splines method is a promising alternative for numerically analyzing the mechanical structure performance of LPG cylinders, particularly in energy storage issues. Full article

This paper presents a novel adaptive crack-tip extended isogeometric analysis (adaptive CT XIGA) framework based on locally refined B-splines (LR B-splines) for efficient and accurate fracture modeling in two-dimensional solids. The XIGA method facilitates crack modeling without requiring the specific locations of crack faces and enables crack propagation simulation without remeshing by employing localized enrichment functions. LR B-splines, as an advanced extension of B-splines and NURBS, offer high-order continuity, precise geometric representation, and local refinement capabilities, thereby enhancing computational accuracy and efficiency. Various local mesh refinement strategies, designed based on crack and crack-tip locations, are investigated. Among these strategies, the crack-tip topological refinement strategy is adopted for local refinement in the adaptive CT XIGA framework. Stress intensity factors (SIFs) are evaluated using the contour interaction integral technique, while the maximum circumferential stress criterion is adopted to predict the crack growth direction. Numerical examples demonstrate the accuracy, efficiency, and robustness of adaptive CT XIGA. The results confirm that the proposed framework achieves superior error convergence rates and significantly reduces computational costs compared to a-posteriori-error-based adaptive XIGA methods, particularly in crack propagation simulations. These advantages establish adaptive CT XIGA as a powerful and efficient tool for addressing complex fracture problems in solid mechanics. Full article

This paper shows that at a certain time-point in the analysis procedure, the accuracy of T-spline based isogeometric analysis (IGA) may be substantially improved by increasing the multiplicity of the inner knots up to the polynomial degree. This task can be performed by considering the Bézier extraction operator matrix elementwise, and thus an increased number of updated control points are easily received in the geometrical and computational models. Nevertheless, after the determination of the unique control points, the Bézier elements near the T-junctions may not be well shaped, and thus minor automatic interventions are required to ensure full (i.e., C⁰ and G⁰) compatibility. The improved IGA-based solution may be used as a reference to determine the a posteriori error estimations in the T-spline elements of the domain, and thus may be a useful tool for IGA adaptation. The methodology is shown in BVPs dominated by Laplace–Poisson equations in rectangular and curvilinear domains, while eigenvalues and eigenvectors were extracted in a rectangular acoustic cavity. Full article

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L₂ error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article

In this paper, we extend our previous work on the dynamic buckling analysis of isogeometric shell structures to the stochastic situation where an isogeometric deterministic dynamic buckling analysis method is combined with spectral-based stochastic modeling of geometric imperfections. To be specific, a modified Generalized-α time integration scheme combined with a nonlinear isogeometric Kirchhoff–Love shell element is used to simulate the buckling and post-buckling problems of cylindrical shell structures. Additionally, geometric imperfections are constructed based on NURBS surface fitting, which can be naturally incorporated into the isogeometric analysis framework due to its seamless CAD/CAE integration feature. For stochastic analysis, the method of separation is adopted to model the stochastic geometric imperfections of cylindrical shells based on a set of measurements. We tested the accuracy and convergence properties of the proposed method with a cylindrical shell example, where measured geometric imperfections were incorporated. The ABAQUS reference solutions are also presented to demonstrate the superiority of the inherited smooth and high-order continuous properties of the isogeometric approach. For stochastic dynamic buckling analysis, we evaluated the buckling load variability and reliability functions of the cylindrical shell with 500 samples generated based on seven nominally identical shells reported in the geometric imperfection data bank. It is noted that the buckling load variability in the cylindrical shell obtained with static nonlinear analysis is also presented to show the differences between dynamic and static buckling analysis. Full article