International Journal For Numerical Methods in Engineering

Gültekin, Osman; Moeineddin, Ahmad; Cansız, Barış; Sveric, Krunoslav; Linke, Axel; Kaliske, Michael

doi: 10.1002/nme.70067pmid: N/A

In this article, we present a model of Physics‐informed Neural Networks (PINNs) for predicting the anisotropic hyperelastic behavior of the human passive myocardium. PINNs adhere to the governing equations and the boundary conditions by integrating physical laws into the neural network architecture. They are used for forward and inverse simulations under non‐standard, complex geometries and loading conditions. The first example features a plane strain shear test, a common protocol in soft tissue mechanics, where we provide a comprehensive comparison of three different total loss functions—namely, the minimization of the PDEs, the total potential energy, or a combination of both—for forward problems as a surrogate to finite element analysis (FEA). The second example deals with a patient‐specific geometry of basal myocardium—obtained from cardiac magnetic resonance imaging—for forward and inverse analyses. Key findings reveal that apart from the accurately predicted primary fields, that is, displacements, the inverse design also provides a true estimate of the anisotropic material parameters from ground truth data obtained from experiments or FEA. Limitations remain in the performance of PINNs for forward simulations of the 2D basal myocardium, particularly with respect to computational demands and sensitivity to network architecture and hyperparameters. Despite challenges in accurately predicting secondary fields, for example, stresses, PINNs demonstrate their potential for inverse simulations, particularly in identifying anisotropic constitutive parameters that can be used in the case of noisy or incomplete datasets in future biomechanical applications.

Mirzaei, Saeed; Hejazi, Mehrdad; Ansari, Reza

doi: 10.1002/nme.70069pmid: N/A

In this study, the bending, buckling, and free vibration analysis of bi‐directional functionally graded porous microbeams with variable thickness are investigated. By utilizing the modified strain gradient theory (MSGT) in conjunction with a sinusoidal shear deformation theory, governing equations are derived using Hamilton's principle within the framework of the non‐uniform rational B‐spline (NURBS)‐based isogeometric analysis (IGA). In addition, the C2‐continuity requirement can be easily achieved by increasing the order of the NURBS basis functions. The MLSPs and the material properties of microbeams vary along with both thickness and axial directions based on the rule of mixture scheme. To consider the effects of porosity, two even and uneven distributions are considered. After verifying the accuracy of the presented approach, the influence of the aspect ratio, gradient indices, different boundary conditions, porosity parameters, variable MLSPs, and thickness on the bending, buckling, and free vibration characteristics of microbeams are investigated.

Seltmann, Stephanie; Hasse, Alexander

doi: 10.1002/nme.70084pmid: N/A

The synthesis of compliant mechanisms (CMs) is frequently achieved through topology optimization. Many synthesis approaches simplify implementation by assuming small distortions, but this limits their practical application since CMs typically undergo large deformations that include geometric and material nonlinearities. CMs designed to generate a desired deformation path at the output points under specific loads are known as path‐generating CMs. However, these CMs face significant challenges in topology optimization, resulting in the development of only a few optimization methods. Existing approaches often include only certain load cases in the optimization process. Consequently, if a CM designed this way encounters different load cases in practice, its path‐generating behavior cannot be guaranteed. The authors have previously contributed to the development of an approach suitable for synthesizing load case‐insensitive CMs. This paper extends that approach to account for nonlinearities, enabling the synthesis of path‐generating CMs. The effectiveness of this extended approach is demonstrated through appropriate design examples. Additionally, the paper presents, for the first time, a shape‐adaptive path‐generating CM.

Müller, Andreas

doi: 10.1002/nme.70088pmid: N/A

Existing methods for constructing splines and Bézier curves on a Lie group G$$ G $$ involve repeated products of exponentials deduced from local geodesics, w.r.t. a Riemannian metric, or rely on general polynomials. Moreover, each of these local curves is supposed to start at the identity of G$$ G $$. Both assumptions may not reflect the actual curve to be interpolated. This paper pursues a different approach to construct splines on G$$ G $$. Local curves are expressed as solutions of the Poisson equation on G$$ G $$. Therewith, the local interpolations satisfies the boundary conditions while respecting the geometry of G$$ G $$. A kth$$ k\mathrm{th} $$‐order approximation of the solutions gives rise to a kth$$ k\mathrm{th} $$‐order product of exponential (POE) spline. Algorithms for constructing 3rd‐ and 4th‐order splines are derived from closed form expressions for the approximate solutions. Additionally, spline algorithms are introduced that allow prescribing a vector field the curve must follow at the interpolation points. It is shown that the established algorithms, where kth$$ k\mathrm{th} $$‐order POE‐splines are constructed by concatenating local curves starting at the identity, cannot exactly reconstruct a kth$$ k\mathrm{th} $$‐order motion. To tackle this issue, the formulations are extended by allowing for local curves between arbitrary points, rather than curves emanating from the identity. This gives rise to a global kth$$ k\mathrm{th} $$‐order spline with arbitrary initial conditions. Several examples are presented, in particular the shape reconstruction of slender rods modeled as geometrically nonlinear Cosserat rods.

Ferrari, Federico; Sigmund, Ole

doi: 10.1002/nme.70082pmid: N/A

Asymptotic post‐buckling theory is applied to sizing and topology optimization of trusses and frames, exploring its potential and current computational difficulties. We show that the designs' post‐buckling response can be controlled by including the lowest two asymptotic coefficients, representing the initial post‐buckling slope and curvature, in the optimization formulation. This also reduces the imperfection sensitivity of the optimized design. The asymptotic expansion can further be used to approximate the structural nonlinear response, and then to optimize for a given measure of the nonlinear mechanical performance, such as, e.g., end‐compliance or complementary work. Examples of linear and nonlinear compliance minimization of trusses and frames show the effective use of the asymptotic method for including post‐buckling constraints in structural optimization.

Pulikkathodi, Afsal; Chamoin, Ludovic; Lacazedieu, Elisabeth; Ramirez, Juan Pedro Berro; Rota, Laurent; Zarroug, Malek

doi: 10.1002/nme.70086pmid: N/A

The article tackles the challenge of effective modeling and simulation of large mechanical structures exhibiting numerous local complex behaviors, as encountered with spot welds in automotive crash numerical analysis. To address this challenge, we propose a nonintrusive local/global coupling strategy, where the local model is a neural network‐based reduced model, specifically a physics‐guided neural network (PGANN). This multiscale strategy enables accurate modeling of complex localized behaviors while maintaining computational efficiency, without modifying the global solver. The proposed approach is validated through a series of structural examples, including full 3D industrial structures with multiple spot welds.

Hu, Jiayi; Zhang, Weisheng; Tang, Shaoqiang

doi: 10.1002/nme.70061pmid: N/A

In this paper, we propose a transformed tensor decomposition method for topology optimization. A transform is performed on the density variable to manipulate its range. The transformed variable is then decomposed as the sum of a number of modes, each in a variable‐separated form. In this way, the number of design variables in discrete form and the optimization time used in each iteration are considerably reduced. Numerical tests are performed to illustrate the nice features of the proposed method with evidence for solving the problems of checkerboard and mesh‐dependency.

doi: 10.1002/nme.7524pmid: N/A

Luo, Guangdong; Zhou, Xiaoping; Wang, Guilin

doi: 10.1002/nme.70055pmid: N/A

Under complex environmental conditions, geomaterial can fail in numerous ways, such as tensile failure, shear failure, and mixed tensile‐shear failure. The first two are extreme cases of the latter. To address this issue, a phase field material point method enhanced by modified B‐K criteria is proposed to handle mixed tensile‐shear failure. Unlike the finite element method (FEM), the governing equations of the coupled‐field system are defined on material points and solved using bilinear interpolation functions on a regular background grid. The material points are allowed to move flexibly within the background grid, making it easier to accurately track the failure zone. A standard iterative staggered algorithm is utilized to solve the coupled‐field system. The effectiveness and reliability of this approach are validated through a set of representative examples.

Manguri, Ahmed; Magisano, Domenico; Jankowski, Robert

doi: 10.1002/nme.70096pmid: N/A

This paper presents an effective and robust computational method of gradient‐based methodology for weight minimization of geometrically nonlinear structures, considering 3D trusses as exemplary case study. The optimization framework can accommodate multiple different constraints: (i) bounds on the cross‐sectional area of each design element, (ii) prescribed ranges for displacements and stresses, and (iii) nonlinear stability for geometries such as arches and domes. For large structures, this results in numerous optimization variables and constraints, including the highly nonlinear (ii) and (iii). Such constraints are evaluated consistently and simultaneously by combining path‐following finite element analysis and null vector method. Typically, the gradient of the nonlinear structural response is approximated numerically, which is computationally intensive and can introduce inaccuracies deteriorating the optimization process. In contrast, this work derives a fully analytical gradient evaluation for nonlinear deformation, stress, and stability constraints. This is implemented directly within the finite element solver, enhancing robustness and computational efficiency of the optimization. Validation examples range from simple benchmarks to large structures.