GAMM Mitteilungen最新文献

A stress equilibration procedure for hyperelastic material models is proposed and analyzed in this paper. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs, in a vertex-patch-wise manner, an H(div)-conforming approximation to the first Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy stress is weakly symmetric in the sense that its antisymmetric part is zero tested against continuous piecewise linear functions. Our main result is the identification of the subspace of test functions perpendicular to the range of the local equilibration system on each patch which turn out to be rigid body modes associated with the current configuration. Momentum balance properties are investigated analytically and numerically and the resulting stress reconstruction is shown to provide improved results for surface traction forces by computational experiments.

Gradient elasticity formulations have the advantage of avoiding geometry-induced singularities and corresponding mesh dependent finite element solution as apparent in classical elasticity formulations. Moreover, through the gradient enrichment the modeling of a scale-dependent constitutive behavior becomes possible. In order to remain C⁰ continuity, three-field mixed formulations can be used. Since so far in the literature these only appear in the small strain framework, in this contribution formulations within the general finite strain hyperelastic setting are investigated. In addition to that, an investigation of the inf sup condition is conducted and unveils a lack of existence of a stable solution with respect to the L²-H¹-setting of the continuous formulation independent of the constitutive model. To investigate this further, various discretizations are analyzed and tested in numerical experiments. For several approximation spaces, which at first glance seem to be natural choices, further stability issues are uncovered. For some discretizations however, numerical experiments in the finite strain setting show convergence to the correct solution despite the stability issues of the continuous formulation. This gives motivation for further investigation of this circumstance in future research. Supplementary numerical results unveil the ability to avoid singularities, which would appear with classical elasticity formulations.

In this contribution, we present a diffuse modeling approach to embed material interfaces into nonconforming meshes with a focus on linear elasticity. For this purpose, a regularized indicator function is employed that describes the distribution of the different materials by a scalar value. The material in the resulting diffuse interface region is redefined in terms of this indicator function and recomputed by a homogenization of the adjacent material parameters. The applied homogenization method fulfills the kinematic compatibility across the interface and the static equilibrium at the interface. In addition, an hℓ-adaptive refinement strategy based on truncated hierarchical B-spline is applied to provide an appropriate and efficient approximation of the diffuse interface region. We justify mathematically and demonstrate numerically that the applied approach leads to optimal convergence rates in the far field for one-dimensional problems. A two-dimensional example illustrates that the application of the hℓ-adaptive refinement strategy allows for a clear reduction of the error in the near and far field and a good resolution of the local stress and strain fields at the interface. The use of a higher continuous B-spline basis leads to efficient computations due to the higher continuity of the diffuse interface model.

In this paper, the use of hp-basis functions with higher differentiability properties is discussed in the context of the finite cell method and numerical simulations on complex geometries. For this purpose, C^k hp-basis functions based on classical B-splines and a new approach for the construction of C¹ hp-basis functions with minimal local support are introduced. Both approaches allow for hanging nodes, whereas the new C¹ approach also includes varying polynomial degrees. The properties of the hp-basis functions are studied in several numerical experiments, in which a linear elastic problem with some singularities is discretized with adaptive refinements. Furthermore, the application of the C^k hp-basis functions based on B-splines is investigated in the context of nonlinear material models, namely hyperelasticity and elastoplasicity with finite strains.

Phase-field approaches to fracture are gaining popularity to compute a priori unknown crack paths. In this work the sensitivity of such phase-field approaches with respect to its model specific parameters, that is, the critical length of regularization, the degradation function and the mobility, is investigated. The susceptibility of the computed cracks to the setting of these parameters is studied for problems of linear and finite elasticity. Furthermore, the convergence properties of different solution strategies are analyzed. Monolithic and staggered solution schemes for the solution of the arising nonlinear discrete systems are studied in detail. To conclude, we demonstrate the versatility of the phase-field fracture approach in a real-world problem by comparing different simulations of conchoidal fracture using structured and unstructured meshes.

In this work, we consider adaptive mesh refinement for a monolithic phase-field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase-field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual-type error estimator for the phase-field fracture model in each time-step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle-point system has three unknowns: displacements, phase-field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

In recent years computational models have become more important for simulating hepatic processes and investigating liver diseases in silico and so various liver models have been published. The complex behavior of biological tissue with its hierarchical structure as well as the blood perfusion through the organ have been described using different approaches and numerical techniques. This paper shows and compares numerical approaches for function and perfusion simulation recently published and compares them with a multiscale function-perfusion model using the extended theory of porous media. We focus on the description of blood perfusion and liver tissue, but also on the simulation of liver diseases or the zonation of processes in the liver. Furthermore, the selected geometry is taken into account.

Motivated by the increasing interest of the biomechanical community towards the employment of strain-gradient theories for solving biological problems, we study the growth and remodeling of a biological tissue on the basis of a strain-gradient formulation of remodeling. Our scope is to evaluate the impact of such an approach on the principal physical quantities that determine the growth of the tissue. For our purposes, we assume that remodeling is characterized by a coarse and a fine length scale and, taking inspiration from a work by Anand, Aslan, and Chester, we introduce a kinematic variable that resolves the fine scale inhomogeneities induced by remodeling. With respect to this variable, a strain-gradient framework of remodeling is developed. We adopt this formulation in order to investigate how a tumor tissue grows and how it remodels in response to growth. In particular, we focus on a type of remodeling that manifests itself in two different, but complementary, ways: on the one hand, it finds its expression in a stress-induced reorganization of the adhesion bonds among the tumor cells, and, on the other hand, it leads to a change of shape of the cells and of the tissue, which is generally not recovered when external loads are removed. To address this situation, we resort to a generalized Bilby-Kröner-Lee decomposition of the deformation gradient tensor. We test our model on a benchmark problem taken from the literature, which we rephrase in two ways: microscale remodeling is disregarded in the first case, and accounted for in the second one. Finally, we compare and discuss the obtained numerical results.

Skeletal muscle is one of the most fascinating and crucial ingredients of motion generation in nature. Since the beginning of science, people dedicate their life as researchers to enhance knowledge about this biological motor. Thus, the scientific knowledge about the skeletal muscle is overwhelmingly broad and detailed. This contribution collects knowledge about the active and passive dynamics of skeletal muscle. Furthermore, it highlights a special perspective in which not only the muscle itself, but also the role muscles play in the interaction with other structures is studied. The first section introduces this systems biophysics perspective, which clusters the investigation of the relations, interactions and dependencies between muscles and the other structures in the movement apparatus. In the second section, the muscles are considered in more detail by describing three approaches to muscle modeling. The third section deals with recent advances based on Hill-type models, such as, for example, the integration of mass.