We generate a basis of the space of bicubic and biquartic C1-smooth geometrically continuous isogeometric functions on bilinear multi-patch domains Ω⊂R2. The basis functions are obtained by suitably ...combining C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains (cf. 18). They are described by simple explicit formulas for their spline coefficients.
These C1-smooth isogeometric functions possess potential for applications in isogeometric analysis, which is demonstrated by several examples (such as the biharmonic equation). In particular, the numerical results indicate optimal approximation power.
•Construction of a basis for bicubic and biquartic C1-smooth isogeometric functions on planar bilinear multi-patch domains.•The basis functions are described by simple explicit formulas for their spline coefficients.•Numerical experiments (e.g. solving the biharmonic equation) showed optimal rates of convergence.
Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into ...a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. In this manuscript, through a self-contained Matlab® implementation, we present an introduction to IGA applied to simple analysis problems and the related computer implementation aspects. Furthermore, implementation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. We also describe the use of IGA in the context of strong-form (collocation) formulations, which has been an area of research interest due to the potential for significant efficiency gains offered by these methods. The code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bézier extraction concept that allows the FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA.
Cellular structures have recently been received a tremendous growth in discussions and applications in the engineering due to their several fascinating structural properties, such as the ...ultra-lightweight, high stiffness and crashworthiness. However, the discussions on the design of cellular structures in the complex structural domain with the non-conforming mesh are still unavailable, resulting from the fact that the non-conforming mesh causes several difficulties in numerical analysis and design optimization. Hence, the main purpose of the work is to propose a Multi-Patch Isogeometric Topology Optimization (MP-ITO) method with powerful capabilities for periodic or graded cellular structures. Firstly, the Nitsche’s method is applied to couple non-conforming meshes in multiple NURBS patches and conduct multi-patch isogeometric analysis. Secondly, a multi-patch topology description model is developed, in which a local smoothing mechanism and the two-resolution scheme of discretization meshes are constructed to avoid terrible structural features and improve smoothness and continuity of boundaries at the interfaces within adjacent subdomains. The separation and independency of the Density Distribution Function (DDF) at each subdomain can offer high flexibility for cellular designs with the imposing of several kinds of periodic constraints. Thirdly, the MP-ITO method is proposed for complex structures and the mathematical formulation for cellular designs considering periodic constraints is developed. Finally, the effectiveness and indispensability of the local smoothing mechanism and the two-resolution scheme in the MP-ITO are discussed, and several numerical examples are addressed to present the compelling effectiveness and capabilities of the MP-ITO method with the high flexibility on the designs of periodic and graded cellular structures.
Summary
Phase‐field modeling, which introduces the regularized representation of sharp crack topologies, provides a convenient strategy for tackling 3D fracture problems. In this work, an adaptive ...isogeometric‐meshfree approach is developed for the phase‐field modeling of brittle fracture in a 3D polycrystalline material. The isogeometric‐meshfree approach uses moving least‐squares approximations to construct the equivalence between isogeometric basis functions and meshfree shape functions, thus inheriting the flexible local mesh refinement scheme from a meshfree method. This refinement scheme is improved by introducing an error estimator that includes both the phase field and its gradient. With the present approach, numerical implementations of the adaptive phase‐field modeling that introduces the anisotropy of fracture resistance in polycrystals are proposed. In this way, propagating cracks can be dynamically tracked, and the mesh near cracks is refined in a meshfree manner without requiring a priori knowledge of crack paths. Furthermore, the intergranular and transgranular crack propagation patterns in polycrystalline materials can be simulated by the present approach. A series of numerical examples that deal with the isotropic and anisotropic fracture are investigated to demonstrate the robustness and effectiveness of the proposed approach.
Accurate numerical modeling of multifield piezoelectric materials is challenging because of the inherent electro-mechanical coupling effect and material anisotropic behaviors. The modeling becomes ...even more difficult especially for problems with non-smooth solutions like crack under dynamic loading. We present in this paper an extension of the extended isogeometric analysis (XIGA) for simulation of two-dimensional fracture mechanics problems in piezoelectric materials under dynamic and static coupled electromechanical loads. The discretization of problem domain is based on basis functions generated from NURBS, which are used for both geometric description and approximation of solution field variables. To capture the discontinuity across the crack-faces and the singularity at the crack-tip, the isogeometric approximation is locally enriched by discontinuous Heaviside function and asymptotic crack-tip branch functions. The sixfold enrichment functions particularly suitable for electromechanical crack-tip singularity of piezoelectric materials are used. To evaluate the generalized fracture parameters, a domain-form of electromechanical interaction integral is employed. For dynamic analysis, the implicit time integration scheme considering inertial effect is taken. Five numerical examples for single and mixed-modes of impermeable cracks are considered and the generalized fracture parameters under dynamic and static loads are analyzed. The accuracy and effectiveness of the proposed XIGA are illustrated through numerical investigations of the generalized dynamic and static fracture parameters. Numerical results are validated against the reference solutions derived from the boundary element methods. The effects of some numerical aspect ratios on generalized fracture parameters are also investigated. Additionally, we present some numerical results of quasi-static crack propagation in piezoelectric solids using the developed XIGA, taking fracture toughness anisotropy of polarized electroelastic materials into account, and employing the maximum modified hoop stress intensity factor criterion for predicting the growing direction of crack.
•XIGA dynamic and static fracture formulation in piezoelectricity is developed.•Crack growth modeling in brittle piezoelectric solids is presented.•Static generalized intensity factors of cracked piezoelectric solids are analyzed.•Transient dynamic responses of impermeable and permeable cracks are studied.•Effects of polarization, enrichments, meshes, loadings, etc. on GIFs are investigated.
We propose an immersed approach for the isogeometric collocation method, combined with the Galerkin-based finite cell method, to avoid the subdivision of complex geometries in too many patches.
The ...presented technology further develops the hybrid collocation concept to accommodate both numerical methods within a single framework, providing a systematic technique for selecting the method to be used.
We perform several numerical tests to demonstrate that the methodology guarantees the same convergence rates obtained using the standard isogeometric collocation method.
Summary
In this paper, a new method is proposed that extend the classical deterministic isogeometric analysis (IGA) into a probabilistic analytical framework in order to evaluate the uncertainty in ...shape and aim to investigate a possible extension of IGA in the field of computational stochastic mechanics. Stochastic IGA (SIGA) method for uncertainty in shape is developed by employing the geometric characteristics of the non‐uniform rational basis spline and the probability characteristics of polynomial chaos expansions (PCE). The proposed method can accurately and freely evaluate problems of uncertainty in shape caused by deformation of the structural model. Additionally, we use the intrusive formulation approach to incorporate PCE into the IGA framework, and the C++ programming language to implement this analysis procedure. To verify the validity and applicability of the proposed method, two numerical examples are presented. The validity and accuracy of the results are assessed by comparing them to the results obtained by Monte Carlo simulation based on the IGA algorithm.
We present a fully explicit dynamic formulation for geometrically exact shear-deformable beams. The starting point of this work is an existing isogeometric collocation (IGA-C) formulation which is ...explicit in the strict sense of the time integration algorithm, but still requires a system matrix inversion due to the use of a consistent mass matrix. Moreover, in that work, the efficiency was also limited by an iterative solution scheme needed due to the presence of a nonlinear term in the time-discretized rotational balance equation. In the present paper, we address these limitations and propose a novel fully explicit formulation able to preserve high-order accuracy in space. This is done by extending a predictor–multicorrector approach, originally proposed for standard elastodynamics, to the case of the rotational dynamics of geometrically exact beams. The procedure relies on decoupling the Neumann boundary conditions and on a rearrangement and rescaling of the mass matrix. We demonstrate that an additional gain in terms of computational cost is obtained by properly removing the angular velocity-dependent nonlinear term in the rotational balance equation without any significant loss in terms of accuracy. The high-order spatial accuracy and the improved efficiency of the proposed formulation compared to the existing one are demonstrated through some numerical experiments covering different combinations of boundary conditions.
•We compare isogeometric collocation (IGA-C) with isogeometric Galerkin and FEA.•Of particular interest are quadrature cost and accuracy vs. computing time.•IGA-C has the potential to offer a more ...efficient alternative to existing technology.•Motivated by the two-scale relation of B-splines we introduce the concept of weighted collocation.•Its combination with hierarchical refinement of NURBS leads to efficient and robust adaptive IGA-C.
We compare isogeometric collocation with isogeometric Galerkin and standard C0 finite element methods with respect to the cost of forming the matrix and residual vector, the cost of direct and iterative solvers, the accuracy versus degrees of freedom and the accuracy versus computing time. On this basis, we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric Galerkin and standard C0 finite element methods, when a specified level of accuracy is to be achieved with minimum computational cost. We then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of NURBS basis functions and collocation points derived from the corresponding multi-level Greville abscissae. We introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two-scale relation of B-splines. Using weighted collocation in the transition regions between hierarchical levels, we are able to reliably handle coincident collocation points that naturally occur for multi-level Greville abscissae. The resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency, local adaptivity, robustness and straightforward implementation, which we illustrate by numerical examples in one, two and three dimensions.
We present an extension of isogeometric collocation to coupled cardiac electromechanical problems. We develop a staggered solution scheme that takes advantage of isogeometric collocation to reduce ...the computational effort in the simulation of the mechanical step, guaranteeing high accuracy for all field variables.
We mainly focus on (i) the strategy adopted to couple the electrical and mechanical sub-problems, (ii) the possibility of handling different meshes to better represent the spatial scales, (iii) and the mitigation of volumetric locking. To this end, we propose a suitable mixed formulation for finite elasticity.
Several numerical tests demonstrate that the mixed formulation retrieves the expected convergence rates under h-refinement and the effectiveness of the proposed solution scheme for electromechanics.
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