International Journal for Numerical Methods in Engineering

Satheesh, Abhiroop; Schmidt, Christoph P.; Wall, Wolfgang A.; Meier, Christoph

doi: 10.1002/nme.7373pmid: N/A

Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data T=RQTΛQ$$ \boldsymbol{T}=\boldsymbol{R}{\boldsymbol{Q}}^T\kern0.00em \boldsymbol{\Lambda} \boldsymbol{Q} $$, followed by an individual interpolation of the two rotation tensors R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ and the positive definite diagonal eigenvalue tensor Λ$$ \boldsymbol{\Lambda} $$ resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group 𝕊𝕆(3), either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, R$$ \boldsymbol{R} $$ and Q$$ \boldsymbol{Q} $$ remain orthogonal tensors and Λ$$ \boldsymbol{\Lambda} $$ remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

Raze, Ghislain; Volvert, Martin; Kerschen, Gaetan

doi: 10.1002/nme.7376pmid: N/A

This work proposes a novel efficient method to track the evolution of amplitude extrema featured by frequency responses of nonlinear systems using the harmonic balance method. Means to compute the amplitude of a Fourier series are first outlined, and a set of equations characterizing a local extremum of a nonlinear frequency response amplitude curve is derived. Efficient numerical procedures are used to evaluate these equations and their derivatives (including second‐order ones) to embed them in a predictor‐corrector continuation framework. The proposed approach is illustrated on three examples of increasing complexity, namely a Helmholtz–Duffing oscillator, a two‐degree‐of‐freedom system with a modal interaction, and a doubly clamped von Kàrmàn beam with a nonlinear tuned vibration absorber.

Wang, Gradey; Darve, Eric; Lew, Adrian J.

doi: 10.1002/nme.7360pmid: N/A

Laser Powder Bed Fusion (LPBF) is a form of metal additive manufacturing in which a laser traces a path on a metal powder bed to heat powder particles and progressively melt and/or fuse them together to form a part. Outcomes of LPBF are functions of the metal's temperature history and are strongly affected by the path traced by the laser. Using temperature field optimization for LPBF as a motivating application, we focus on a class of combinatorial problems where admissible control policies are a permutation of a set of control actions. We abstract this class of problems as an optimal control problem with the objective of identifying an admissible control policy that minimizes a cost function of a dynamical system's state. While, in principle, the effect of the control actions on the cost function can last an infinitely long time, we will consider systems in which the effects of these control actions can be considered to last short (finite) periods of time. In this paper, we formalize this class of combinatorial problems as a Traveling Salesperson Problem with History and prove its equivalence to an Equality Generalized Traveling Salesperson Problem (E‐GTSP), enabling the use of well‐developed E‐GTSP solvers. We demonstrate this equivalence by computing the solutions obtained using an E‐GTSP solver for an LPBF‐inspired application.

Shi, A.; Persson, P.‐O.

doi: 10.1002/nme.7379pmid: N/A

Mesh optimization procedures are generally a combination of node smoothing and discrete operations which affect a small number of elements to improve the quality of the overall mesh. These procedures are useful as a post‐processing step in mesh generation procedures and in applications such as fluid simulations with severely deforming domains. In order to perform high‐order mesh optimization, these ingredients must also be extended to high‐order (curved) meshes. In this work, we present a method to perform local element operations on curved meshes. The mesh operations discussed in this work are edge/face swaps, edge collapses, and edge splitting (more generally refinement) for triangular and tetrahedral meshes. These local operations are performed by first identifying the patch of elements which contain the edge/face being acted on, performing the operation as a “straight‐sided one" by placing the high‐order nodes via an isoparametric mapping from the master element, and smoothing the high‐order nodes on the elements in the patch by minimizing a Jacobian‐based high‐order mesh distortion measure. Since the initial “straight‐sided guess” from the placement of the nodes via the isoparametric mapping frequently results in invalid elements, the distortion measure must be regularized which allows for mesh untangling for the optimization to succeed. We present several examples in 2D and 3D to demonstrate these local operations and how they can be combined with a high‐order node smoothing procedure to maintain mesh quality when faced with severe deformations.

Nikolopoulos, Stefanos; Kalogeris, Ioannis; Stavroulakis, George; Papadopoulos, Vissarion

doi: 10.1002/nme.7372pmid: N/A

Recent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost‐efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the ‘exact’ solution of the problem. To address this issue, the present work proposes the use of up‐to‐date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large‐scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD‐2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD‐2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

doi: 10.1002/nme.7264pmid: N/A

Abdelhamid, Mohamed; Czekanski, Aleksander

doi: 10.1002/nme.7368pmid: N/A

This work extends force coupling in the topology optimization of fluid‐structure interaction problems from hydrostatic to total stresses through the inclusion of viscous stress components. The superconvergent patch recovery technique is implemented to remove the discontinuities in velocity derivatives over the finite elements boundaries. The sensitivity analysis is derived analytically for the superconvergent patch recovery approach and further verified through the use of the complex‐step derivative approximation method. Numerical examples demonstrate a differentiation in the optimized designs using pressure versus total stress coupling depending on the flow characteristics of the design problem.

Han, Jike; Hirayama, Daigo; Shintaku, Yuichi; Moriguchi, Shuji; Terada, Kenjiro

doi: 10.1002/nme.7371pmid: N/A

This study presents an enhancement of the diffusive‐discrete crack transition scheme (10.1002/nme.7169) to describe dynamic fracture at finite strain. In the enhanced scheme, the crack initiation, propagation, and bifurcation processes are determined from an energy minimization problem based on crack phase‐field theory, and the predicted diffusive crack is replaced by the discrete representation using the finite cover method. In the meantime, numerical damping is introduced to maintain computational stability and avoid distortion of the physical mesh in the finite cover context. By taking advantage of the features of the diffusive‐discrete crack transition scheme, the proposed approach enables us to stably simulate a series of dynamic fracture events involving crack initiation at an arbitrary location, propagation, and bifurcation in arbitrary directions, arbitrary divisions of an original object into multiple portions, and independent motions of divided portions. After spatial and temporal discretizations by the finite cover method and the Newmark method are described, as well as the simulation algorithm of the enhanced finite cover‐based staggered iterative procedure for dynamic fracture, several representative numerical examples are presented to demonstrate the performance and capabilities of the developed approach.

Bellet, Michel; Keumo Tematio, Joël; Zhang, Yancheng

doi: 10.1002/nme.7378pmid: N/A

To reduce computational time of thermo‐mechanical simulation of additive manufacturing processes, the inherent strain (IS) method is quite efficient, but suffers from a lack of predictivity, preventing concurrent quantitative prediction of distortion and stress. In order to investigate the predictive ability of the IS method, a new variant of the method is proposed in which the determination of the IS tensor is based on simulation results from a standard thermo‐elastic‐viscoplastic simulation applied to a few learning layers of the studied part. More precisely, the inherent strains are determined by a direct resolution performed in each finite element, with a negligible computation cost. Because of these specific features, the predictive ability of this IS method is formulated as its ability to replicate the distortion and stress predicted by the reference standard thermo‐elastic‐viscoplastic simulation. The proposed IS method provides perfect results in a validation test where full‐field inherent strains are used in each added layer: the distortion and stress predicted by the thermo‐elastic‐viscoplastic simulation are exactly replicated by the IS method. However, when applying the IS method to the simulation of an entire part, here a turbine blade mock‐up, the results are degraded with respect to the reference solution obtained by the thermo‐elastic‐viscoplastic simulation. Results are then discussed to evaluate the limitations of the proposed method, and of the IS method in general.

Yang, Zhenjun; Zhang, Xin; Wang, Zhenyu; Li, Q. M.

doi: 10.1002/nme.7370pmid: N/A

A 3D mesoscale finite element modelling approach is developed for simulating complicated damage and fracture behaviour in steel fibre‐reinforced concrete (SFRC) with explicit modelling of fibre–matrix interfaces. In this approach, a new 3D four‐noded cohesive‐frictional coupled interface element is developed to model the nonlinear interfacial bond‐slip behaviour, supplemented by a kinematic multiple‐point‐constraint (kMPC) algorithm to simulate the wrapping effect of the mortar around the fibres. They are implemented as a user‐defined element (UEL) and a user‐defined MPC subroutine in ABAQUS, respectively. Three cohesive‐frictional constitutive relationships are proposed to describe the nonlinear bond‐slip behaviour of different fibre–matrix compositions. The new approach is validated by single fibre pullout tests, direct tensile tests and three‐point bending tests of SFRC specimens with randomly distributed fibres. The results show that the new approach is capable of effectively capturing typical failure mechanisms in SFRC, such as fibre yielding, matrix failure, and fibre–matrix debonding and slipping.