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In the present work the geometrically exact beam model is formulated in terms of unit quaternions. A projection‐based discretization approach is proposed which is based on a normalization of the quaternion approximation. The discretization relies on NURBS shape functions and, alternatively, on Lagrangian interpolation. The redundancy of the quaternions is resolved by applying the method of Lagrange multipliers. In a second step the Lagrange multipliers are eliminated circumventing the need to solve saddle point systems. The resulting finite elements retain the objectivity of the underlying beam formulation. Optimal rates of convergence are observed in representative numerical examples. [ABSTRACT FROM AUTHOR]

Summary: In this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis. [ABSTRACT FROM AUTHOR]

Summary: We investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B‐splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B‐spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary‐fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B‐splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B‐splines, even in the boundary‐fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B‐spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one‐dimensional elastic‐plastic bar impacting a rigid wall is illustrative. [ABSTRACT FROM AUTHOR]

A collocation method is developed for discrete fracture models in the context of the partition‐of‐unity method. Spline technologies used in isogeometric analysis (IGA) are exploited to provide a smooth inter‐element transition of gradients, thus allowing to get rid of extra flux terms at element boundaries which are generated by Lagrange polynomials. Bézier extraction is utilised to formulate IGA commensurate with a standard finite element data‐structure. The efficacy of the proposed approach is examined through different numerical examples and is compared with other discrete methods for fracture analysis. The proposed approach is competitive in terms of accuracy with the least computational cost, rendering it a suitable candidate for superseding available collocation approaches for fracture simulation. Moreover, the approach naturally assesses the possibility of physics informed neural networks for fracture simulation, to which collocation is central. [ABSTRACT FROM AUTHOR]

We propose a new class of high‐order time‐marching schemes with dissipation control and unconditional stability for parabolic equations. High‐order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized‐α$$ \alpha $$ method delivers unconditional stability and second‐order accuracy in time and controls the numerical dissipation in the discrete spectrum's high‐frequency region. We extend the generalized‐α$$ \alpha $$ methodology to obtain high‐order time marching methods with high accuracy and dissipation control in the discrete high‐frequency range. Furthermore, we maintain the original stability region of the second‐order generalized‐α$$ \alpha $$ method in the new higher‐order methods; we increase the accuracy of the generalized‐α$$ \alpha $$ method while keeping the unconditional stability and user‐control features on the high‐frequency numerical dissipation. The methodology solves k>1,k∈ℕ$$ k>1,k\in \mathbb{N} $$ matrix problems and updates the system unknowns, which correspond to higher‐order terms in Taylor expansions to obtain (3/2k)th$$ \left(3/2k\right)\mathrm{th} $$‐order method for even k$$ k $$ and (3/2k+1/2)th$$ \left(3/2k+1/2\right)\mathrm{th} $$‐order for odd k$$ k $$. A single parameter ρ∞$$ {\rho}^{\infty } $$ controls the high‐frequency dissipation, while the update procedure follows the formulation of the original second‐order method. Additionally, we show that our method is A‐stable, and for ρ∞=0$$ {\rho}^{\infty }=0 $$ we obtain an L‐stable method. Furthermore, we extend this strategy to analyze the accuracy order of a generic method. Lastly, we provide numerical examples that validate our analysis of the method and demonstrate its performance. First, we simulate heat propagation; then, we analyze nonlinear problems, such as the Swift–Hohenberg and Cahn–Hilliard phase‐field models. To conclude, we compare the method to Runge–Kutta techniques in simulating the Lorenz system. [ABSTRACT FROM AUTHOR]

Consistent treatment of large rotations in common Reissner–Mindlin formulations is a complicated task. Reissner–Mindlin formulations that use a hierarchic parametrization provide an elegant way to facilitate large rotation shell analyses. This can be achieved by the assumption of linearized transverse shear strains, resulting in an additive split of strain components, which technically simplifies implementation of corresponding shell finite elements. The present study aims at validating this assumption by systematically comparing numerical solutions with those of a newly implemented hierarchic and fully nonlinear Reissner–Mindlin shell element. [ABSTRACT FROM AUTHOR]

Summary: This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness‐to‐weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine‐scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so‐called inexact FETI‐DP method where the local, cell‐wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell‐wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro‐scale data, thereby making our method also multiscale and matrix‐free. The solver is tested against various 2D and 3D analyses. It shows major gains compared to black‐box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes. [ABSTRACT FROM AUTHOR]

Porous structures by additive manufacturing have fascinating and compelling performance compared with solid structures. The stress‐related porous infill designs which could greatly mitigate the effects of the intrinsic high residual stress in additive manufacturing process have gained increasing attention. In the current work, we propose a promising Isogeometric Topology Optimization (ITO) method for porous infill structures with stress minimization to avoid the occurrence of stress concentrations in additive manufacturing. The IsoGeometric Analysis (IGA) and induced p‐norm aggregation are utilized to develop a stress‐minimization topology description model for infill design, which can remove the mesh dependency and offer benefits for improving numerical accuracy and convergence stability. We also introduced global volume constraints to easily control the usage of material and eliminate the over‐fine structures affecting the printing accuracy. Several numerical examples are performed to demonstrate the effectiveness and advantages of the proposed ITO method on porous infill designs with stress minimization. The laser powder bed fusion (LPBF) technique is employed to fabricate several prototypes, and the performance are evaluated by experiments. The advancements of our work are demonstrated effectively, which is adapted for additive manufacturing and practical application. [ABSTRACT FROM AUTHOR]

Based on the explicit stiffness computation formula of NURBS (non‐uniform rational B‐splines) element and tensor product decomposed explicit filter, a novel multi‐material isogeometric topology optimization (MMITO) method is put forward to generate multi‐material structures efficiently. The explicit NURBS element stiffness computation formula is: derived from the Bėzier extraction operator under the combination of scaled and rotational geometric transformations between NURBS elements, and the stiffness matrices of NURBS elements can be computed by the stiffness matrix of a Bėzier element without resorting to the time‐consuming integral procedure. As the presented MMITO is a nested optimization framework, we apply the explicit filtering technique to update the multi‐phase design variables efficiently in combination with the tensor product decomposition method. With the decomposed weight matrices of the filter and the explicit form of the NURBS stiffness matrix, we obtain the equivalent but much simplified preprocessing procedure and sensitivity analysis for the MMITO method. A comprehensive set of multi‐material numerical examples are optimized to verify that, while improving the preprocessing efficiency and reducing the memory overhead, the present MMITO scheme is able to generate semblable two‐dimensional and three‐dimensional multi‐material structures with identical structural performance to the traditional scheme. Therefore, the proposed MMITO method is a promising approach to optimizing multi‐material structures. [ABSTRACT FROM AUTHOR]

This study presents the incorporation of the effective gradient‐free proportional topology optimization algorithm into the framework of isogeometric analysis. The minimization of the compliance is considered, and the solid isotropic material with penalization method is used. The geometry, displacements, and density are all described by non‐uniform rational B‐spline (NURBS) basis functions. The density at an integration point is determined proportionally to its compliance. Then, the NURBS description of the density is constructed elementwise by deriving a relation between densities assigned to integration points and control points. The global NURBS description of the density for the whole domain is a blend of those from elements. Furthermore, a multiresolution scheme is presented by means of k$$ k $$‐refinement technique to enable the efficient performance for large‐scale problems. The accuracy and efficiency of the proposed approach are assessed through six numerical examples, including two‐ and three‐dimensional structures, with several rigorous tests and comparisons with the gradient‐based optimality criteria algorithm. [ABSTRACT FROM AUTHOR]

Continuous‐assumed‐strain (CAS) elements were recently introduced (Casquero and Golestanian. Comput Methods Appl Mech Eng. 2022; 399:115354.) to remove the membrane locking present in quadratic C1$$ {C}^1 $$‐continuous NURBS‐based discretizations of linear plane curved Kirchhoff rods. In this work, we generalize CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane curved Timoshenko rods. CAS elements are an assumed strain treatment that interpolates the shear and membrane strains at the knots using linear Lagrange polynomials. Consequently, the inter‐element continuity of the shear and membrane strains is maintained. The numerical experiments considered in this work show that CAS elements excise the spurious oscillations in shear and membrane forces caused by shear and membrane locking. Furthermore, when using CAS elements with either full or reduced integration, the convergence of displacements, rotations, and stress resultants is independent of the slenderness ratio up to 104$$ 1{0}^4 $$ while the convergence is highly dependent on the slenderness ratio when using NURBS elements. We apply the locking treatment of CAS elements to quadratic C0$$ {C}^0 $$‐continuous NURBS and the resulting element type is named discontinuous‐assumed‐strain (DAS) elements. Comparisons among CAS and DAS elements show that once locking is properly removed, C1$$ {C}^1 $$ continuity across element boundaries results in higher accuracy than C0$$ {C}^0 $$ continuity across element boundaries. Lastly, CAS elements result in a simple numerical scheme that does not add any significant computational burden in comparison with the locking‐prone NURBS‐based discretization of the Galerkin method. [ABSTRACT FROM AUTHOR]

This work proposes a moving morphable component (MMC)‐based topology optimization approach for designing graded lattice flexoelectric nanostructures, which are generally considered to have better flexoelectric properties. To this end, the infill structure is described with the use of a set of components. A coordinate perturbation is made to the topology description functions of the components to achieve a graded distribution of the flexoelectric materials. Resorting to the explicit parameters of the components, the singular disconnected structure can be naturally controlled with the use of lower/upper bound constraints. To ensure the continuity in the approximation of the PDEs of flexoelectricity, an ersatz material model is developed by combining isogeometric analysis (IGA) with MMC. Since MMC provides definite feature sizes in the optimization result, the complicated infill lattice can be manufactured by additive manufacturing directly. The flexoelectric performances of the design results from the proposed approach are demonstrated through both numerical examples and experimental measurements. It is shown that the obtained novel infill lattice structures have better flexoelectric performance when compared with traditional structural configurations. [ABSTRACT FROM AUTHOR]

Summary: An enhancement of the extended B‐spline‐based implicit material point method (EBS‐MPM) is developed to avoid pressure oscillation and volumetric locking. The EBS‐MPM is a stable implicit MPM that enables the imposition of arbitrary boundary conditions thanks to the higher‐order EBS basis functions and the help of Nitsche's method. In particular, by means of the higher‐order EBS basis functions, the EBS‐MPM can suppress the cell‐crossing errors caused by material points crossing the background grid boundaries and can avoid both the stress oscillations arising from inaccurate numerical integration and the ill‐conditioning of the resulting tangent matrices. Although the higher‐order EBS basis functions are known to avoid volumetric locking, the problem of pressure oscillation has not yet been resolved. Therefore, to suppress pressure oscillation due to quasi‐incompressibility, we propose the incorporation of the F‐bar projection method into the EBS‐MPM, which is compatible with the higher‐order EBS basis functions. Three representative numerical examples are presented to demonstrate the capability of the proposed method in suppressing both the pressure oscillation and volumetric locking. The results of the proposed method are compared to those of the finite element method with F‐bar elements and those of isogeometric analysis with quadratic NURBS elements. [ABSTRACT FROM AUTHOR]

Cellular materials, in particular metal or polymer foams and beam lattices, are characterized by a complex architecture where the material is concentrated in slender struts. This study is concerned with the construction of numerically efficient sample‐specific digital twin for the analysis and characterization of the mechanical behavior of such meta‐materials. A method to build an explicit, light, and analysis‐suitable isogeometric beam model directly from digital images is proposed. More precisely, as a main contribution, the virtual image correlation method is extended to the case of multiple interconnected beams with varying cross‐sections. Special care is also given to the initialization for better efficiency and robustness. Synthetic and real examples of increasing complexity, that is, curved beams, lattices, and foams, are presented to account for the performance of the developed workflow from digital images to an isogeometric twin for mechanical modeling and simulation. [ABSTRACT FROM AUTHOR]

In continuation of our previous work on nonlinear stability analysis of trimmed isogeometric thin shells, this contribution is an extension to dynamic buckling analyses for predicting reliably complex snap‐through and mode‐jumping behavior. Specifically, a modified generalized‐α$$ \alpha $$ time integration scheme is used and combined with a nonlinear isogeometric Kirchhoff–Love shell element to provide second‐order accuracy while introducing controllable high‐frequency dissipation. In addition, a weak enforcement of essential boundary conditions based on a penalty approach is considered with a particular focus on the inhomogeneous case of imposed prescribed displacements. Moreover, we propose a least‐squares B‐spline surface fitting approach and corresponding error measures to model both eigenmode based and measured geometric imperfections. The imperfect geometries thus obtained can be naturally integrated into the framework of isogeometric nonlinear dynamic shell analysis. Based on this idea, the different modeling methods and the influence of the appropriately considered geometric imperfections on the dynamic buckling behavior can be investigated systematically. Both perfect and geometrically imperfect shell models are considered to assess the performance of the proposed method. We compare our method with established developments in this field and demonstrate superior achievements with regard to solution quality and robustness. [ABSTRACT FROM AUTHOR]

Powell‐Sabin B‐splines are employed to model progressive fracturing in a fluid‐saturated porous medium. These splines are defined on triangles and are 풞1‐continuous throughout the domain, including the crack tips, so that crack initiation can be evaluated directly at the tip. On one hand, the method captures stresses and fluid fluxes more accurately than when using standard Lagrange elements, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. On the other hand, the method avoids limitations for discrete crack analysis which adhere to isogeometric analysis. A crack is introduced directly in the physical domain. Due to the use of triangles, remeshing and crack path tracking are straightforward. During remeshing transfer of state vectors (displacement, fluid pressure) is required from the old to the new mesh. The transfer is done using a new approach which exploits a least‐squares fit with the energy balance and conservation of mass as constraints. The versatility and accuracy to simulate free crack propagation are assessed for mode‐I and mixed‐mode fracture problems. [ABSTRACT FROM AUTHOR]

This work presents a numerical formulation to model isotropic viscoelastic material behavior for membranes and thin shells. The surface and the shell theory are formulated within a curvilinear coordinate system, which allows the representation of general surfaces and deformations. The kinematics follow from Kirchhoff–Love theory and the discretization makes use of isogeometric shape functions. A multiplicative split of the surface deformation gradient is employed, such that an intermediate surface configuration is introduced. The surface metric and curvature of this intermediate configuration follow from the solution of nonlinear evolution laws—ordinary differential equations—that stem from a generalized viscoelastic solid model. The evolution laws are integrated numerically with the implicit Euler scheme and linearized within the Newton–Raphson scheme of the nonlinear finite element framework. The implementation of membrane and bending viscosity is verified with the help of analytical solutions and shows ideal convergence behavior. The chosen numerical examples capture large deformations and typical viscoelasticity behavior, such as creep, relaxation, and strain rate dependence. It is also shown that the proposed formulation can be straightforwardly applied to model boundary viscoelasticity of 3D bodies. [ABSTRACT FROM AUTHOR]

The computation cost of matrix formation in isogeometric analysis can be drastically reduced by employing tensor decomposition, but the performance like accuracy and robustness still depends on the approach used to realize the low‐rank approximation for the nonpolynomial integral kernels. In the buckling analysis of laminated shells with solid‐shell model, the integral kernels are not smooth in the thickness direction due to their material and stress distribution and have a high complexity because the curvilinear coordinate systems are involved in the representation of anisotropic constitutive relations, which makes it more difficult to guarantee the decomposition accuracy. This paper proposes a robust high‐fidelity tensor‐decomposition based matrix formation method for stiffness matrix and geometric stiffness matrix of the isogeometric buckling analysis problem. The proposed method avoids using a spline approximation and employs the hierarchical block‐wise decomposition approach (HBD) to obtain a determinate decomposition result, which saves more time without loss of accuracy and produces less canonical ranks with a reliable procedure. Besides, the matrix calculation is partly independent of the number of layers, showing a higher efficiency when the number of layers is larger. Finally, several experiments with various Gaussian curvatures are implemented to validate the proposed method. [ABSTRACT FROM AUTHOR]

In this work, an object‐oriented geometric engine is proposed to solve problems with discontinuities, for instance, material interfaces and cracks, by means of unfitted, immersed, or enriched finite element methods (FEMs). Both explicit and implicit representations, such as geometric entities and level sets, are introduced to describe configurations of discontinuities. The geometric engine is designed in an object‐oriented way and consists of several modules. For efficiency, a k‐d tree data structure that partitions the background mesh is constructed for detecting cut elements whose neighbors are found by means of a dual graph structure. Moreover, the implementation for creating enriched nodes, integration elements, and physical groups is described in detail, and the corresponding pseudo‐code is also provided. The complexity and efficiency of the geometric engine are investigated by solving 2‐D and 3‐D discontinuous models. The capability of the geometric engine is demonstrated on several numerical examples. Topology optimization and problems with intersecting discontinuities are handled with enriched FEMs, where enriched discretizations obtained from the geometric engine are used for the analysis. Furthermore, polycrystalline structures that overlap with an unfitted mesh are considered, where integration elements are created so they align with grain boundaries. Another example shows that the Stanford bunny, which is discretized by a surface mesh with triangular elements, can be fully immersed into a 3‐D background mesh. Finally, we share a list of main findings and conclude that the proposed geometric engine is general, robust, and efficient. [ABSTRACT FROM AUTHOR]

An isogeometric Galerkin approach for analysing the free vibrations of piezoelectric shells is presented. The shell kinematics is specialized to infinitesimal deformations and follow the Kirchhoff–Love hypothesis. Both the geometry and physical fields are discretized using Catmull–Clark subdivision bases. This provides the required C1$$ {C}^1 $$‐continuous discretization for the Kirchhoff–Love theory. The crystalline structure of piezoelectric materials is described using an anisotropic constitutive relation. Hamilton's variational principle is applied to the dynamic analysis to derive the weak form of the governing equations. The coupled eigenvalue problem is formulated by considering the problem of harmonic vibration in the absence of external load. The formulation for the purely elastic case is verified using a spherical thin shell benchmark. Thereafter, the piezoelectric shell formulation is verified using a one dimensional piezoelectric beam. The piezoelectric effect and vibration modes of a transverse isotropic curved plate are analyzed and evaluated for the Scordelis–Lo roof problem. Finally, the eigenvalue analysis of a CAD model of a piezoelectric speaker shell structure showcases the ability of the proposed method to handle complex geometries. [ABSTRACT FROM AUTHOR]

In this article, a new isogeometrical level set topology optimization is introduced. In previous studies, the level set function is approximated by b‐splines basis functions and the control net is updated during the optimization process. Since control points are not basically on the level set surface, a discrepancy between zero level of function that represents boundaries of the structure, and zero level of its control netis appeared. In this article, the idea is solving level set equation (LSE) over the parameter space of b‐splines basis functions defined in the IGA model. Afterwards, the updated level set function is mapped into the physical space. The level set function is approximated over a grid defined in parameter space which is constantly a unit square. By doing so, control net of the IGA model and the grid for solving LSE are separated. Two well‐known radial basis functions (RBF) and reaction–diffusion (RD) based methods are employed to solve the LSE. The proposed method is applied to minimize the mean compliance when certain amount of material is used and also for weight minimization subject to local stress constraints. Several numerical examples are presented to demonstrate performance and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents a general, nonlinear isogeometric finite element formulation for rotation‐free shells with embedded fibers that captures anisotropy in stretching, shearing, twisting, and bending—both in‐plane and out‐of‐plane. These capabilities allow for the simulation of large sheets of heterogeneous and fibrous materials either with or without matrix, such as textiles, composites, and pantographic structures. The work is a computational extension of our earlier theoretical work that extends existing Kirchhoff‐Love shell theory to incorporate the in‐plane bending resistance of initially straight or curved fibers. The formulation requires only displacement degrees‐of‐freedom to capture all mentioned modes of deformation. To this end, isogeometric shape functions are used in order to satisfy the required C1$$ {C}^1 $$‐continuity for bending across element boundaries. The proposed formulation can admit a wide range of material models, such as surface hyperelasticity that does not require any explicit thickness integration. To deal with possible material instability due to fiber compression, a stabilization scheme is added. Several benchmark examples are used to demonstrate the robustness and accuracy of the proposed computational formulation. [ABSTRACT FROM AUTHOR]

We introduce a novel computational framework for the multiscale simulation of higher‐order continua that allows for the consideration of first‐, second‐, and third‐order effects at both micro‐ and macro‐level. In line with classical two‐scale approaches, we describe the microstructure via representative volume elements that are attached at each integration point of the macroscopic problem. To take account of the extended continuity requirements of independent fields at micro‐ and macro‐level, we discretize both scales via isogeometric analysis (IGA). As a result, we obtain an IGA2‐method that is conceptually similar to the well‐known FE2‐method. We demonstrate the functionality and accuracy of this novel multiscale method by means of a series of multiscale simulations involving different kinds of higher‐order continua. [ABSTRACT FROM AUTHOR]

The integration of generalized differential quadrature techniques and finite element (FE) methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi‐patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two‐dimensional solution domain of the plate model into sub‐domains, called patches, and then to apply the VDQ method along with the FE mapping technique for each patch. The formulation is presented in a weak form and due to the C1‐continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher‐order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher‐order plate model. [ABSTRACT FROM AUTHOR]

An extended isogeometric analysis (XIGA) approach is proposed for modeling fracturing in a fluid‐saturated porous material. XIGA provides a definition of the discontinuity independent of the underlying mesh layout, which obviates the need of knowing the crack extension direction a priori. Unlike Lagrange shape functions used in the standard finite element approach, non‐uniform rational B‐splines (NURBS) provide a higher‐order interelement continuity which leads to a continuous fluid flow also at element boundaries, thereby satisfying the local mass balance. It also leads to an improved estimate of the crack path due to a smoother stress distribution. The NURBS basis functions are cast in finite element data structure using Bézier extraction. To model the discontinuity, the Heaviside sign function is utilized within the displacement and the pressure fields, complemented by the shifting and the blending techniques to enforce compatibility perpendicular and parallel to the crack path, respectively. Different aspects of the approach are assessed through examples comprising straight and curved crack paths for stationary and propagating discontinuities. [ABSTRACT FROM AUTHOR]

This contribution presents an isogeometric approach in three dimensions for nonlinear problems in solid mechanics and structural dynamics. The proposed approach aligns with the boundary representation modeling technique in CAD. Isogeometric analysis is combined with the parameterization of the scaled boundary finite element method. In this way, the boundary description of 3D solids in CAD can be directly utilized for the analysis in an isogeometric framework. The approximation of the solution on the boundary is based on bi‐variate NURBS. We also approximate the interior of the solid with uni‐variate B‐splines to facilitate the solution of nonlinear problems. The nonlinear case is extended to three dimensions in this present work. The method is further extended to dynamic problems. The mass and damping matrices are derived using the same basis functions. The solution of the entire solid is obtained with the Galerkin method. We study several nonlinear and dynamic problems with simple geometries and arbitrary number of boundaries. Moreover, a fiber reinforced composite serves as demonstration for complex shapes. [ABSTRACT FROM AUTHOR]

Lattice‐skin structures composed of a thin‐shell skin and a lattice infill are widespread in nature and large‐scale engineering due to their efficiency and exceptional mechanical properties. Recent advances in additive manufacturing, or 3D printing, make it possible to create lattice‐skin structures of almost any size with arbitrary shape and geometric complexity. We propose a novel gradient‐based approach to optimizing both the shape and infill of lattice‐skin structures to improve their efficiency further. The respective gradients are computed by fully considering the lattice‐skin coupling while the lattice topology and shape optimization problems are solved in a sequential manner. The shell is modeled as a Kirchhoff–Love shell and analyzed using isogeometric subdivision surfaces, whereas the lattice is modeled as a pin‐jointed truss. The lattice consists of many cells, possibly of different sizes, with each containing a small number of struts. We propose a penalization approach akin to the SIMP (solid isotropic material with penalization) method for topology optimization of the lattice. Furthermore, a corresponding sensitivity filter and a lattice extraction technique are introduced to ensure the stability of the optimization process and to eliminate scattered struts of small cross‐sectional areas. The developed topology optimization technique is suitable for nonperiodic, nonuniform lattices. For shape optimization of both the shell and the lattice, the geometry of the lattice‐skin structure is parameterized using the free‐form deformation technique. The topology and shape optimization problems are solved in an iterative, sequential manner. The effectiveness of the proposed approach and the influence of different algorithmic parameters are demonstrated with several numerical examples. [ABSTRACT FROM AUTHOR]

We present a new isogeometric analysis (IGA) approach based on extended Loop subdivision scheme for solving various geometric flows defined on subdivision surfaces. The studied flows include the second‐order, fourth‐order, and sixth‐order geometric flows, such as averaged mean curvature flow, constant mean curvature flow, and minimal mean‐curvature‐variation flow, which are generally derived by minimizing the associate energy functionals with L2‐gradient flow respectively. The geometric flows are discretized by means of subdivision based IGA, where the finite element space is formulated by the limit form of the extended Loop subdivision for different initial control meshes. The basis functions, consisting of quartic box‐splines corresponding to each subdivided control mesh, are utilized to represent the geometry exactly. For the cases of the evolution of open surfaces with any shape boundary, high‐order continuous boundary conditions derived from the mixed variational forms of the geometric flows should be implemented to be consistent with the isogeometric concept. For time discretization, we adopt an adaptive semi‐implicit Euler scheme. By several numerical experiments, we study the convergence behaviors of the proposed approach for solving the geometric flows with high‐order boundary conditions. Moreover, the numerical results also show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The multi‐modal Koiter method is a reduction technique for estimating quickly the nonlinear buckling response of structures under mechanical loads requiring a fine discretization. The reduced model is based on a quadratic approximation of the full model using a few linear buckling modes and their second order corrections, followed by the projection of the equilibrium equations onto the modal subspace. In this article, the method is reformulated for geometrically nonlinear thermoelastic analyses of shell structures. The starting point is an isogeometric solid‐shell discretization with an accurate modeling of thermal strains, temperature‐dependent materials and general temperature distributions. The equilibrium path is defined as a displacement versus temperature amplifier curve, while mechanical loads are kept constant. The strain energy nonlinearity with respect to the temperature amplifier is the main obstacle in the definition of an accurate reduced model. This task is achieved by coherent asymptotic expansions in mixed form using independent stress variables at the integration points and accurate linear buckling modes obtained by a two‐point mixed eigenvalue problem. Structures made of isotropic and multi‐layered composite materials, including variable angle tow laminates, are considered to show the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents the Bézier extraction based isogeometric approach to multi-objective topology optimization of periodic microstructures. The approach utilizes the B-splines based Bézier elements as the finite element representation at both macro and micro levels. The equivalent elastic properties of the representative volume element (RVE) are computed by the numerical homogenization method with the periodic boundary conditions on the control points (CPs) of the B-spline mesh. The multi-objective optimization problems including the material bulk modulusmaximization, negative Poisson's ratio, and concurrent topology optimization of the composite structures are formulated by using the Bézier elements based isogeometric analysis. The distance-based density distribution on the CPs of the B-splinemesh is proposed as the initialization for designing the RVE, which is more robust than the uniform density distribution towards the optimal results. Several numerical examples are presented to illustrate the effectiveness of the proposed approach, and a variety of isotropic and anisotropic RVEs and composite structures are obtained. Meanwhile, various influences on the optimal design of the RVE and macrostructure are also discussed. [ABSTRACT FROM AUTHOR]

An extended approach is developed by blending isogeometric analysis and the first‐order local maximum entropy for the standard and the enhanced fields, respectively. Isogeometric analysis facilitates the accurate parametrization of the geometry in general, particularly the exact geometric parametrization of the conic curves and quadratic surfaces using NURBS. On the other hand, the local maximum entropy leads to an improved estimate for the enhanced part due to its infinite continuity. Moreover, local maximum entropy paves the way to a nonelementwise crack propagation owing to its meshfree characteristic. To enforce compatibility, the shifting technique is amended for the meshfree enhanced part to localize the effect of the Heaviside function to a narrow region around the crack. Next, a blending technique is exploited to remove the effect of the discontinuity in front of the crack tip. The viability of the approach is illustrated at the hand of several examples comprising straight and curved crack propagation. [ABSTRACT FROM AUTHOR]

In the present work, we propose an adaptive topology optimization (TO) method under suitably graded truncated hierarchical B‐spline refinement and coarsening (SGTHB‐TO). A series of algorithms have been devised to implement admissible adaptivity of SGTHB‐TO by resorting to the definitions of suitably graded isogeometric hierarchical meshes. We apply the proposed SGTHB‐TO method to two‐dimensional and three‐dimensional TO problems of compliance and compliant mechanism. According to the numerical results, SGTHB‐TOs with or without admissible restriction share similar optimization processes, except that numerical artifact, such as edge blurring and islanding gray regions occurring in SGTHB‐TO without admissible restriction, can be eliminated by SGTHB‐TO with admissible parameter equaling to 2. Besides, the convergence rate can be largely accelerated by imposing an admissible constraint on SGTHB‐TO for TO problems with geometric nonlinearity existing in the design domain, which is attributed to the improved numerical accuracy by the admissible adaptive discretization scheme of isogeometric analysis (IGA). Therefore, the SGTHB‐TO method with the admissible parameter equaling to 2 is a more effective adaptive TO method than SGTHB‐TO without considering admissible restriction, in terms of both numerical stability and convergence rate. [ABSTRACT FROM AUTHOR]

Summary: We present an adaptive isogeometric‐analysis approach to elasto‐capillary fluid‐solid interaction (FSI), based on a diffuse‐interface model for the binary fluid and an Arbitrary‐Lagrangian‐Eulerian formulation for the FSI problem. We consider approximations constructed from adaptive high‐regularity truncated hierarchical splines, as employed in the isogeometric analysis (IGA) paradigm. The considered adaptive strategy comprises a two‐level hierarchical a posteriori error estimate. The hierarchical a posteriori error estimate directs a support‐based refinement procedure. To attain robustness of the solution procedure for the aggregated binary‐fluid‐solid‐interaction problem, we apply a fully monolithic solution procedure and we introduce a continuation process in which the diffuse interface of the binary fluid is artificially enlarged on the coarsest levels of the adaptive‐refinement procedure. To assess the capability of the presented adaptive IGA method for elasto‐capillary FSI, we conduct numerical computations for a configuration pertaining to a sessile droplet on a soft solid substrate. [ABSTRACT FROM AUTHOR]

This article presents an enhanced version of our previous work, hybrid nonuniform subdivision (HNUS) surfaces, to achieve optimal convergence rates in isogeometric analysis (IGA). We introduce a parameter λ (14<λ<1) to control the rate of shrinkage of irregular regions, so the method is called tuned hybrid nonuniform subdivision (tHNUS). Thus, HUNS is a special case of tHNUS when λ=12. While introducing λ in hybrid subdivision significantly complicates the theoretical proof of G1 continuity around extraordinary vertices, reducing λ can recover optimal convergence rates when tHNUS functions are used as a basis in IGA. From the geometric point of view, tHNUS retains comparable shape quality as HNUS under nonuniform parameterization. Its basis functions are refinable and the geometric mapping stays invariant during refinement. Moreover, we prove that a tHNUS surface is globally G1‐continuous. From the analysis point of view, tHNUS basis functions form a nonnegative partition of unity, are globally linearly independent, and their spline spaces are nested. In the end, we numerically demonstrate that tHNUS basis functions can achieve optimal convergence rates for the Poisson's problem with nonuniform parameterization around extraordinary vertices. [ABSTRACT FROM AUTHOR]

One of the most appealing techniques in local refinement for isogeometric analysis (IGA) is hierarchical B‐splines which can be improved by resorting to truncation mechanism called truncated hierarchical B‐splines. Although it has a simple concept, it involves implementing the hierarchical definition of shape functions to existing codes. In this contribution, we present a simple method defined by knot vectors of different levels of hierarchical meshes resulted from knot insertion techniques. In fact, the proposed method is a generalization of the classical Bézier extraction with an element viewpoint which enhances the procedure of adaptive finite element analysis in error estimation based on marking of the elements. This results in an adaptive IGA based on multilevel extraction to the hierarchical overlay of functions. The convergence and computational efficiency of the proposed method are demonstrated through benchmark examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Summary: The surface energy a phase‐field approach to brittle fracture in anisotropic materials is also anisotropic and gives rise to second‐order gradients in the phase field entering the energy functional. This necessitates C1 continuity of the basis functions which are used to interpolate the phase field. The basis functions which are employed in isogeometric analysis (IGA), such as nonuniform rational B‐splines and T‐splines naturally possess a higher order continuity and are therefore ideally suited for phase‐field models which are equipped with an anisotropic surface energy. Moreover, the high accuracy of spline discretizations, also relative to their computational demand, significantly reduces the fineness of the required discretization. This holds a fortiori if adaptivity is included. Herein, we present two adaptive refinement schemes in IGA, namely, adaptive local refinement and adaptive hierarchical refinement, for phase‐field simulations of anisotropic brittle fracture. The refinement is carried out using a subdivision operator and exploits the Bézier extraction operator. Illustrative examples are included, which show that the method can simulate highly complex crack patterns such as zigzag crack propagation. An excellent agreement is obtained between the solutions from global refinement and adaptive refinement, with a reasonable reduction of the computational effort when using adaptivity. [ABSTRACT FROM AUTHOR]

Summary: The objective of this study is to present an extended isogeometric formulation for cohesive fracture. The approach exploits the higher order interelement continuity property of nonuniform rational B‐splines (NURBS), in particular the higher accuracy that results for the stress prediction, which yields an improved estimate for the direction of crack propagation compared to customary Lagrangian interpolations. Shifting is used to ensure compatibility with the surrounding discretization, where, different from extended finite element methods, the affected elements stretch over several rows perpendicular to the crack path. To avoid fine meshes around the crack tip in case of cohesive fracture, a blending function is used in the extension direction of the crack path. To comply with standard finite element data structures, Bézier extraction is used. The absence of the Kronecker‐delta property in the higher order interpolations of isogeometric analysis impedes the enrichment scheme and compatibility enforcement. These issues are studied comprehensively at the hand of several examples, while crack propagation analyses show the viability of the approach. [ABSTRACT FROM AUTHOR]

Benefited from the accuracy improvement in modeling physical problem of complex geometry and integrating the discretization and simulation, the isogeometric analysis in boundary element method (IGABEM) has been drawn a great deal of attention. The nearly singular integrals of 2D potential problem in the IGABEM are addressed by a semianalytical scheme in the present work. We use the subtraction technique to separate the integrals to singular and nonsingular parts, where the singular parts can be calculated by the analytical formulae derived by utilizing a series of integration by parts, while the nonsingular parts are calculated numerically with fewer quadrature points. Comparing the present semianalytical results with the ones of exact solutions, we find that the present method can obtain precise potential and flux densities of inner points much closer to the boundary without refining the elements nearby. Sufficient comparisons with other regularization schemes, such as the exponential and sinh transformation methods, are also conducted. The results in the numerical examples show the competitiveness of the present method, especially when calculating the nearly strongly and highly singular integrals during the simulation of the flux density. [ABSTRACT FROM AUTHOR]

Summary: This contribution is concerned with a coupling approach for nonconforming NURBS patches in the framework of an isogeometric formulation for solids in boundary representation. The boundary representation modeling technique in CAD is the starting point of this approach. We parameterize the solid according to the scaled boundary finite element method and employ NURBS basis functions for the approximation of the solution. Therefore, solid surfaces consist of several sections, which can be regarded as patches and discretized independently. The main objective of this study is to derive an approach for the connection of independent sections in order to allow for local refinement and thus an accurate and efficient discretization of the computational domain. Nonconforming sections are coupled with a mortar approach within a master‐slave framework. The coupling of adjacent sections ensures the equality of mutual deformations along the interface in a weak sense and is enforced by constraining the NURBS basis functions on the interface. We apply this approach to nonlinear problems in two dimensions and compare the results with conforming discretizations. [ABSTRACT FROM AUTHOR]

Summary: A recently proposed phase‐field model for cohesive fracture is examined. Previous investigations have shown stress oscillations to occur when using unstructured meshes. It is now shown that the use of nonuniform rational B‐splines (NURBS) as basis functions rather than traditional Lagrange polynomials significantly reduces this oscillatory behavior. Moreover, there is no effect on the global structural behavior, as evidenced through load‐displacement curves. The phase‐field model imposes restrictions on the interpolation order of the NURBS used for the three different fields: displacement, phase field, and crack opening. This holds within the Bézier element, but also at the boundaries, where a reduction to 𝒞0‐continuity yields optimal results. Application to a range of cases, including debonding of a hard fiber embedded in a soft matrix, illustrates the potential of the cohesive phase‐field model. [ABSTRACT FROM AUTHOR]

Summary: An efficient optimization framework is developed in this study by integrating auxiliary projection‐based multigrid isogeometric reanalysis (MG‐IGR) and metaheuristic searching techniques. It is well known that the inherent characteristics of isogeometric analysis (IGA) are superior in shape optimization problems. Inheriting the characteristics of IGA, an auxiliary projection‐based MG reanalysis (MGR) is proposed to construct mapping between the mesh before modification and after modification during the optimization process. Subsequently, MG‐IGR is utilized to reanalyze the modified design efficiently by reusing the initial evaluated results. Moreover, the proposed MG‐IGR also eliminates the restriction of mesh consistency. In this framework, the structure can be designed directly through parameterized control of the non‐uniform rational B‐spline (NURBS) model, and the MG‐IGR fast solver enables any metaheuristic algorithm to perform the optimization procedure. Moreover, the accuracy of the simulation can be guaranteed by the NURBS model and the convergence criterion of the MG. Finally, two geometric optimization examples are presented to validate the performance of the developed framework. [ABSTRACT FROM AUTHOR]

Summary: In this paper, the impact problem and the subsequent wave propagation are considered. For the contact discretization an intermediate non‐uniform rational B‐spline (NURBS) layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient element‐based integration. The impact event is ill‐posed and requires a regularization to avoid propagating stress oscillations. A nonlinear mesh‐dependent penalty regularization is used, where the stiffness of the penalty regularization increases upon mesh refinement. Explicit time integration methods are well suited for wave propagation problems, but are efficient only for diagonal mass matrices. Using a spectral element discretization in combination with a NURBS contact layer the bulk part of the mass matrix is diagonal. [ABSTRACT FROM AUTHOR]

Summary: This study presents a level set–based topology optimization with isogeometric analysis (IGA) for controlling high‐frequency electromagnetic wave propagation in a domain with periodic microstructures (unit cells). The high‐frequency homogenization method is applied to characterize the macroscopic high‐frequency waves in periodic heterogeneous media whose wavelength is comparative to or smaller than the representative length of a unit cell. B‐spline basis functions are employed for the IGA discretization procedure to improve the performance of electromagnetic wave analysis in a unit cell and topology optimization. Also, to keep the same order of continuity on the periodic boundaries as on other element edges in the domain, we propose the extended domain approach, while incorporating Floquet periodic boundary condition (FPBC). Two types of optimization problems are taken as examples to demonstrate the effectiveness of the proposed method in comparison with the standard finite element analysis (FEA). The optimization results provide optimized topologies of unit cells qualified as anisotropic metamaterials with hyperbolic and bidirectional dispersion properties at the macroscale. [ABSTRACT FROM AUTHOR]

We present a variationally separable splitting technique for the generalized‐α method for solving parabolic partial differential equations. We develop a technique for a tensor‐product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multidimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user‐controlled high‐frequency dissipation while minimizing unwanted low‐frequency dissipation. The method has second‐order accuracy in time and optimal rates (hp+1 in L2 norm of u and hp in L2 norm of ∇u) in space. We present the spectral analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy. [ABSTRACT FROM AUTHOR]

Despite the significant progress that has been made since then in computational models and methods for Fluid-Structure Interaction (FSI), many open challenges still remain, and computational FSI continues to be an active area of investigation and development. An essential complication in elasto-capillary FSI pertains to the modeling of the contact line, that is, the triple point corresponding to the intersection of the fluid meniscus with the fluid-structure interface. Int J Numer Methods Eng. 2021; 122 (19): 5430 - 5449. https://doi.org/10.1002/nme.6372 12 Boncoraglio G, Farhat C, Bou-Mosleh C. Model reduction framework with a new take on active subspaces for optimization problems with linearized fluid-structure interaction constraints. [Extracted from the article]

Summary: We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties. [ABSTRACT FROM AUTHOR]

Summary: A recent distortion‐tolerant unsymmetric 8‐node hexahedral solid‐shell element US‐ATFHS8, which takes the analytical solutions of linear elasticity as the trial functions, is successfully extended to geometric nonlinear analysis. This extension is based on the corotational (CR) approach due to its simplicity and high efficiency, especially for geometric nonlinear analysis where the strain is still small. Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinear analysis, the incremental corotational formulations of the nonlinear solid‐shell element US‐ATFHS8 are derived within the updated Lagrangian (UL) framework, in which an appropriate updated strategy for linear analytical trial functions is proposed. Numerical examples show that the present nonlinear element US‐ATFHS8 possesses excellent performance for various rigorous tests no matter whether regular or distorted mesh is used. Especially, it even performs well in some situations that other conventional elements cannot work. [ABSTRACT FROM AUTHOR]

Summary: This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples. [ABSTRACT FROM AUTHOR]

Summary: A collocation method has been recently developed as a powerful alternative to Galerkin's method in the context of isogeometric analysis, characterized by significantly reduced computational cost, but still guaranteeing higher‐order convergence rates. In this work, we propose a novel adaptive isogeometric analysis meshfree collocation (IGAM‐C) for the two‐dimensional (2D) elasticity and frictional contact problems. The concept of the IGAM‐C method is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree approach, which facilitates the robust mesh adaptivity in isogeometric collocation. The proposed method reconciles collocation at the Greville points via the discretization of the strong form of the equilibrium equations. The adaptive refinement in collocation is guided by the gradient‐based error estimator. Moreover, the resolution of the nonlinear equations governing the contact problem is derived from a strong form to avoid the disadvantages of numerical integration. Numerical examples are presented to demonstrate the effectiveness, robustness, and straightforward implementation of the present method for adaptive analysis. [ABSTRACT FROM AUTHOR]