International Journal for Numerical Methods in Engineering

doi: 10.1002/nme.6219pmid: N/A

Russ, Jonathan B.; Waisman, Haim

doi: 10.1002/nme.6334pmid: N/A

Optimal design of structures for fracture resistance is a challenging subject. This appears to be largely due to the strongly nonlinear governing equations associated with explicitly modeling fracture propagation. We propose a topology optimization formulation, in which low weight structures are obtained with significantly increased resistance to brittle fracture, in which crack propagation is explicitly modeled with the phase field approach. By contrast to our previous work, several important features are included which greatly assist the optimizer in dealing with the strongly discontinuous brittle fracture process, including a new objective function, which provides additional path information to the optimizer. Increased local control of the topology is introduced via a smoothed threshold function in the phase field fracture formulation and a constraint relaxation continuation scheme is proposed to alleviate some difficulty during the initial optimization iterations. The derivation of the analytical, path‐dependent sensitivities for the relevant functions is provided and the results from two benchmark numerical examples are presented which demonstrate the effectiveness of the proposed method.

Liu, Juanjuan; Li, Enying; Wu, Yunqiang; Wang, Hu

doi: 10.1002/nme.6335pmid: N/A

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.

Davydov, Denis ; Pelteret, Jean‐Paul; Arndt, Daniel; Kronbichler, Martin; Steinmann, Paul

doi: 10.1002/nme.6336pmid: N/A

This work investigates matrix‐free algorithms for problems in quasi‐static finite‐strain hyperelasticity. Iterative solvers with matrix‐free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix‐free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix‐free operator and the geometric multigrid preconditioner is compared to the matrix‐based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix‐free methods for finite‐strain solid mechanics are very promising, outperforming linear matrix‐based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.

Remy Bernard Devloo, Philippe ; Durán, Omar; Monteiro Farias, Agnaldo; Maria Gomes, Sônia

doi: 10.1002/nme.6337pmid: N/A

Effects of nonaffine elements on the accuracy of 3D H(div)‐conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart‐Thomas or Nédélec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k−1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n−1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme.

Wu, Hao ; Wu, Pingbo; Xu, Kai; Li, Fansong

doi: 10.1002/nme.6338pmid: N/A

The accuracy of the finite element model is essential for fatigue analysis and vibration of a flexible body. This paper presents a finite element model updating (FEMU) algorithm based on crows search algorithm incorporated with Levy flight (LFCSA). The proposed algorithm is tested on the updating of two different cases: a simple structure (beam) and a complex structure (gearbox housing). To verify the performance of LFCSA, two optimization algorithm, which is the standard CSA and the particle swarm optimization algorithm with Levy flight (LFPSO), are applied to FEMU of the simple beam. The result of the comparison shows that the LFCSA has sufficient convergence speed and global search ability in the implementation of FEMU. In the case of gearbox housing, the error of frequencies and mode shapes are analyzed to indicate that the LFCSA gives a satisfactory result in a complex structure.

Wu, Chi ; Fang, Jianguang; Zhou, Shiwei; Zhang, Zhongpu; Sun, Guangyong; Steven, Grant P.; Li, Qing

doi: 10.1002/nme.6340pmid: N/A

Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level‐set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase‐field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time‐dependent crack propagation problems. In line with diffusive damage of the phase‐field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction‐diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant‐Friedrichs‐Lewy condition. In this article, three numerical examples, namely, a L‐shaped section, a rectangular slab with predefined cracks, and an all‐ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase‐field fracture context.

Ghaffari Motlagh, Yousef ; Borst, René

doi: 10.1002/nme.6341pmid: N/A

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.

Wu, Zijun ; Fan, Fei; Xiao, Renbin; Yu, Lianqing

doi: 10.1002/nme.6342pmid: N/A

This work presents a generalized substructuring‐based topology optimization method for the design hierarchical lattice structures to maximize the first eigenvalue. In this method, the macrostructure is assumed to be composed of substructures with a common artificial lattice geometry pattern. And two different yet connected scales are considered through a lattice geometry feature parameter. The feature parameter, which can control the material distribution of the substructure, determines the relative density of corresponding substructure. Each substructure is condensed into a super‐element to obtain the associated density‐related matrices. A surrogate model using cubic spline interpolation has been particularly built to map the density to stiffness and mass matrices of condensed super‐elements. The derivatives of super‐element matrices to the associated densities can be evaluated efficiently and accurately. Here, an augmented penalized density for this surrogate model is introduced. And the conventional optimality criteria method is selected as updating method of the density design variables. Numerical examples under two lattice patterns of substructures are shown to validate the correctness and superiority of this substructure‐based topology optimization method.

Schoeder, Svenja ; Sticko, Simon; Kreiss, Gunilla; Kronbichler, Martin

doi: 10.1002/nme.6343pmid: N/A

We propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous Galerkin method for spatial discretization and the arbitrary derivative method with local time stepping (LTS) for time integration. The method is based on a cut finite element approach of high order and uses level set functions to describe curved immersed interfaces. We study under which conditions and to what extent small time step sizes balance cut instabilities, which are present especially for high‐order spatial discretizations. This is done by analyzing eigenvalues and critical time steps for representative cuts. If small time steps cannot prevent cut instabilities, stabilization by means of cell agglomeration is applied and its effects are analyzed in combination with local time step sizes. Based on two examples with general cuts, performance gains of the LTS over the global time stepping are evaluated. We find that LTS combined with cell agglomeration is most robust and efficient.