Martin Ruess

Summary: The buckling properties of thin‐walled structures are sensitive to different sources of imperfections, among which the geometric imperfections are of paramount importance. This work contributes to the methodology of shell buckling analysis with respect to the following aspects: first, we propose an isogeometric analysis framework for the buckling analysis of shell structures which naturally eliminates the geometric discretization errors; second, we introduce a parameter‐free Nitsche‐type formulation for thin shells at large deformations that weakly enforces coupling constraints along trimmed boundaries. In combination with the finite cell method, the proposed conceptual modeling and analysis framework is able to handle engineering‐related shell structures; and third, we introduce a NURBS modeling of measured geometric imperfection fields, which is much closer to the true imperfection shape compared to the classically used faceted FE models. We demonstrate with a number of benchmark problems and engineering models that our proposed framework is able to fully compete with established and highly sophisticated finite element formulations but at a significant higher level of accuracy and reliability of the analysis results.

The mechanical response of cells to stimuli tightly couples biochemical and biomechanical processes, which describe fundamental properties such as cell growth and reorientation. Cells interact continuously with their external surroundings, the extracellular matrix (ECM), by establishing a bond between cell and ECM through the formation of focal adhesions. Focal adhesions are made up of integrins, which are mechanosensitive proteins and responsible for the communication between the cell and the ECM. The governing biochemomechanical processes can be modeled by means of a continuum approach considering mechanical and thermodynamic equilibrium to describe cell contractility and focal adhesion growth. The immanent multiphysical character of cell mechanics involves important aspects such as the coupling of fields of different scales and corresponding interface conditions that are sensitive to the solution of the governing numerical problem. These aspects become even more relevant when considering a feedback loop among the multiphysical solutions fields. In this contribution, we consider solution properties and sensitivity aspects of a nonlinear mechanical continuum model for the prognosis of stress fiber growth and reorientation incorporating a mechanosensitive feedback loop. We provide the governing equations of a Hill model–based stress fiber growth, which is coupled to a thermodynamical approach modeling the focal adhesions. Furthermore, a variational formulation including the algebraic equations is derived for staggered and monolithic solution approaches and the reaction‐diffusion equation that models the feedback mechanism. We test both schemes with regard to reliability, accuracy, and numerical efficiency for different model parameters and loading scenarios. We present algorithmic aspects of the considered solution schemes and reveal their robustness with regard to model refinement in space and time and finally provide an assessment of their overall solution performance for multiphysics problems in the context of cell mechanics.

This work presents a numerical discretization technique for solving 3‐dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high‐order fictitious domain approach, is used for the numerical approximation of the solution without a boundary‐conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub‐domain and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchicalhp‐refinement scheme is used to locally refine the FCM meshes to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for nonsmooth problems. A series of numerical experiments with 2‐ and 3‐dimensional benchmark problems is presented, showing that the proposedhp‐refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra‐implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description.

Summary: We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.

Summary: The Koiter‐Newton method had recently demonstrated a superior performance for nonlinear analyses of structures, compared to traditional path‐following strategies. The method follows a predictor‐corrector scheme to trace the entire equilibrium path. During a predictor step, a reduced‐order model is constructed based on Koiter's asymptotic postbuckling theory that is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid‐shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger‐Reissner variational formulation facilitates the reduced‐order model construction omitting an expensive evaluation of the inherent fourth‐order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out‐of‐balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full‐order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems.

Summary: The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.