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This paper presents an effective method to automatically construct trivariate tensor-product spline models of complicated geometry and arbitrary topology. Our method takes as input a solid model defined by its triangulated boundary surface. Using cuboid decomposition, an initial polycube approximating the input boundary mesh is built. This polycube serves as the parametric domain of the tensor-product spline representation required for isogeometric analysis. The polycube’s nodes and arcs decompose the input model’s boundary into quadrilateral patches, and these patches form hexahedral domains. Using aligned global parameterization, the nodes are re-positioned and the arcs are re-routed across the surface in a way to achieve low overall patch distortion, and alignment to principal curvature directions and sharp features. The optimization process is based on one of the main contributions of this paper: a novel way to design cross fields with topological (i.e., imposed singularities) and geometrical (i.e., imposed directions) constraints by solving only sparse linear systems. Based on the optimized polycube and parameterization, compatible B-spline boundary surfaces are reconstructed. Finally, the interior volumetric parameterization is computed using Coon’s interpolation. In the context of parametric studies based on geometrical parameters, this method can be used to compute the morphing required for reduced order modeling. For different parametric instances with the same topology but different geometries, this method allows to have the same representation: i.e., meshes (or parameterizations) with the same topology. The efficiency and the robustness of the proposed approach are illustrated by several examples. [ABSTRACT FROM AUTHOR]

Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

In this paper, we define analysis-suitable++ (AS++) T-splines which include analysis-suitable (AS) T-splines as a special case and maintain all the good mathematical properties as AS T-splines. We prove that AS++ T-splines are always linear independent regardless of the knot values and show that the classical construction of the dual basis for tensor-product B-splines and AS T-splines can be generalized to AS++ T-spline spaces. We also discuss how all of these issues pave the way to a mathematical theory for AS++ T-splines. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate ability of proper generalized decomposition (PGD) to solve transient elastodynamic models in space–time domain. Classical methods use time integration schemes and an incremental resolution process. We propose here to use standard time integration methods in a non-incremental strategy. As a result, PGD gives a separated representation of the space–time solution as a sum of tensorial products of space and time vectors, that we interpret as space–time modes. Recent time integration schemes are based on multi-field formulations. In this case, separated representation can be constructed using state vectors in space and same vectors in time. However, we have experienced bad convergence order using this decomposition. Furthermore, temporal approximation must be the same for all fields. Thus, we propose an extension of classical separated representation for multi-field problems. This multi-field PGD (MF-PGD) uses space and time vectors that are different for each field. Calculation of decomposition is done using a monolithic approach in space and time, potentially allowing the use of different approximations in space and time. Finally, several simulations are performed with the transient elastodynamic problem with one dimension in space. Different approximations in time are investigated: Newmark scheme, single field time discontinuous Galerkin method and two fields time continuous and discontinuous Galerkin methods. [Copyright &y& Elsevier]

Abstract: Modeling large systems usually requires metamodels or response surfaces (RSs) of sub-systems. These metamodels are using sampling points in a parameter domain and related responses provided by the solution of parametric partial differential equations (PDEs). Between sampling points the responses are interpolated. We propose to incorporate the RS approximation into the weak form of parametric partial differential equations (PDEs). Hence a multidimensional model-reduction can be achieved. RSs provide very fast predictions. But their enrichment, by adding new sampling points, requires new response evaluations, and therefore new solutions of PDEs. Although it is off-line computations, their complexity does not facilitates the study of large-scale non-linear problems involving a large number of parameters. Reduced-order models can facilitate the solution of the PDEs used to enrich the RSs. We propose a multidimensional a priori model-reduction method to generate or to enrich RSs. It is coined multidimensional because the fields to forecast are defined over an augmented domain in terms of dimension. They are functions of both space variables and parameters that simultaneously evolve in time. This changes the functional space related to the weak form of the PDEs and the definition of the reduced-bases. It has a significant impact on the proposed model-reduction method. In particular, the variable interpolation in the framework of reduced-basis approximations has to be reconsidered. Moreover, a multidimensional reduced integration domain (MRID) is proposed to reduce the complexity of the reduced formulation. It is a subdomain of the full multidimensional domain. The multidimensional hyper-reduction method extracts from the MRID truncated equilibrium equations, truncated residuals and a truncated error indicator. This work is an extension of the a priori hyper-reduction (APHR) method to parametric PDEs coupled to a design of experiments (DOE) method. In this paper, the outputs of the metamodels are the elastic stiffness and the damping coefficient of non-linear mechanical sub-systems that could be incorporated in the model of aircrafts or cars subjected to vibrations. The proposed method has been designed to account for various recent and ongoing research on mathematical formulation of mechanical constitutive equations in material science. Here, we are using a constitutive model proposed by Qi and Boyce for polymers. The three following issues are addressed: Is the multidimensional APHR method more efficient than the APHR method applied individually on separated simulations? What is the efficiency of the proposed approach when adding sampling points in the current parameter domain? What is the efficiency of the method when adding a new dimension to the parameter domain, i.e. when adding a new parameter to enrich a former parametric study? [Copyright &y& Elsevier]

Abstract: Coupled optimization methods based on multi-objective genetic algorithms and approximation models are widely used in engineering optimizations. In the present paper, a similar framework is proposed for the aerodynamic optimization of turbomachinery by coupling the well known multi-objective genetic algorithm—NSGA-II and back propagation neural network. The verification results of mathematical problems show that the coupled method with the origin NSGA-II cannot get the real Pareto front due to the prediction error of BPNN. A modified crowding distance is proposed in cooperation with a coarse-to-fine approaching strategy based on the iterations between NSGA-II and BPNN. The results of mathematical model problems show the effect of these improving strategies. An industrial application case is implemented on a transonic axial compressor. The optimization objectives are to maximize efficiencies of two working points and to minimize the variation of the choked mass flow. CFD simulation is employed to provide the performance evaluation of initial training samples for BPNN. The optimized results are compared with optimization results of a single objective optimization based on weighting function. The comparison shows that the present framework can provide not only better solutions than the single objective optimization, but also various alternative solutions. The increase of computational costs is acceptable especially when approximation models are used. [Copyright &y& Elsevier]

Abstract: In this paper, four- and eight-node quadrilateral finite element models which can readily be incorporated into the standard finite element program framework are devised for plane Helmholtz problems. In these models, frame (boundary) and domain approximations are defined. The former is obtained by nodal interpolation and the latter is truncated from Trefftz solution sets. The equality of the two approximations are enforced along the element boundary. Both the Bessel and plane wave solutions are employed to construct the domain approximation. For full rankness, a minimal of four and eight domain modes are required for the four- and eight-node elements, respectively. By using local coordinates and directions, rank sufficient and invariant elements with minimal and close to minimal numbers of domain approximation modes are devised. In most tests, the proposed hybrid-Trefftz elements with the same number of nodes yield close solutions. In absolute majority of the tests, the proposed elements are considerably more accurate than their single-field counterparts. [Copyright &y& Elsevier]

Abstract: In this paper the grad–div stabilization for the incompressible Navier–Stokes finite element approximations is considered from two different viewpoints: (i) as a variational multiscale approach for the pressure subgrid modeling and (ii) as a stabilization procedure of least-square type. Some new error estimates for the linearized problem with the grad–div stabilization are proved with the help of norms induced by the pressure Schur complement operator. We discuss the stabilization parameter choice arising in the frameworks of least-square and multiscale methods and consider assumptions which allow to relate both approaches. [Copyright &y& Elsevier]

Abstract: The reliability of computer predictions of physical events depends on several factors: the mathematical model of the event, the numerical approximation of the model, and the random nature of data characterizing the model. This paper addresses the mathematical theories, algorithms, and results aimed at estimating and controlling modeling error, numerical approximation error, and error due to randomness in material coefficients and loads. A posteriori error estimates are derived and applications to problems in solid mechanics are presented. [Copyright &y& Elsevier]