We present a concurrent material and structure optimization framework for multiphase hierarchical systems that relies on homogenization estimates based on continuum micromechanics to account for ...material behavior across many different length scales. We show that the analytical nature of these estimates enables material optimization via a series of inexpensive “discretization-free” constraint optimization problems whose computational cost is independent of the number of hierarchical scales involved. To illustrate the strength of this unique property, we define new benchmark tests with several material scales that for the first time become computationally feasible via our framework. We also outline its potential in engineering applications by reproducing self-optimizing mechanisms in the natural hierarchical system of bamboo culm tissue.
In this paper, we develop a geometrically flexible technique for computational fluid–structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve ...function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term “immersogeometric analysis” to identify this paradigm.
The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid–structure interface traction, arriving at Nitsche’s method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance.
To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.
•We present a method for immersogeometric analysis of flow around complex objects.•The tetrahedral finite cell method resolves boundary geometry accurately.•We enforce Dirichlet boundary conditions ...weakly with Nitsche’s method.•A variational multiscale formulation captures effects of subgrid turbulence.•The methodology is found effective on a 3D benchmark and an industrial problem.
We present a tetrahedral finite cell method for the simulation of incompressible flow around geometrically complex objects. The method immerses such objects into non-boundary-fitted meshes of tetrahedral finite elements and weakly enforces Dirichlet boundary conditions on the objects’ surfaces. Adaptively-refined quadrature rules faithfully capture the flow domain geometry in the discrete problem without modifying the non-boundary-fitted finite element mesh. A variational multiscale formulation provides accuracy and robustness in both laminar and turbulent flow conditions. We assess the accuracy of the method by analyzing the flow around an immersed sphere for a wide range of Reynolds numbers. We show that quantities of interest such as the drag coefficient, Strouhal number and pressure distribution over the sphere are in very good agreement with reference values obtained from standard boundary-fitted approaches. We place particular emphasis on studying the importance of the geometry resolution in intersected elements. Aligning with the immersogeometric concept, our results show that the faithful representation of the geometry in intersected elements is critical for accurate flow analysis. We demonstrate the potential of our proposed method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor.
In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding ...a symmetric positive-definite term to the consistent mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. We perform numerical experiments for the linear wave equation in one and two dimensions, for quadrilateral elements and triangular elements, and for up to fourth order polynomial basis functions. Despite the increase in critical time-step size, we do not observe adverse effects in terms of spatial accuracy and orders of convergence. To extend the method to non-linear problems, we introduce a linear approximation. Our three-dimensional experiments with tetrahedral and hexahedral elements show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.
We describe a strategy for implicit a posteriori error estimation in cut finite elements. Our approach is based on the definition of local residual-driven corrector problems that use a local order ...elevation of the finite element space to construct a correction of the current approximation. The recovered higher-order accurate approximation is then used to construct error estimation in energy norm. We discuss implications of this scheme in the presence of cut elements, for instance regarding the construction of local corrector regions or the imposition of local boundary conditions. Combining the estimator with the finite cell method and a mesh refinement scheme, we numerically demonstrate its effectivity in terms of predicting the true error and its suitability to steer mesh adaptivity. Our results confirm that the estimation achieves the same effectivity in cut meshes as in standard boundary-fitted meshes, irrespective of the polynomial degree.
The finite cell method is an embedded domain method, which combines the fictitious domain approach with higher-order finite elements, adaptive integration, and weak enforcement of unfitted essential ...boundary conditions. Its core idea is to use a simple unfitted structured mesh of higher-order basis functions for the approximation of the solution fields, while the geometry is captured by means of adaptive quadrature points. This eliminates the need for boundary conforming meshes that require time-consuming and error-prone mesh generation procedures, and opens the door for a seamless integration of very complex geometric models into finite element analysis. At the same time, the finite cell method achieves full accuracy, i.e. optimal rates of convergence, when the mesh is refined, and exponential rates of convergence, when the polynomial degree is increased. Due to the flexibility of the quadrature based geometry approximation, the finite cell method can operate with almost any geometric model, ranging from boundary representations in computer aided geometric design to voxel representations obtained from medical imaging technologies. In this review article, we first provide a concise introduction to the basics of the finite cell method. We then summarize recent developments of the technology, with particular emphasis on the research topics in which we have been actively involved. These include the finite cell method with B-spline and NURBS basis functions, the treatment of geometric nonlinearities for large deformation analysis, the weak enforcement of boundary and coupling conditions, and local refinement schemes. We illustrate the capabilities and advantages of the finite cell method with several challenging examples, e.g. the image-based analysis of foam-like structures, the patient-specific analysis of a human femur bone, the analysis of volumetric structures based on CAD boundary representations, and the isogeometric treatment of trimmed NURBS surfaces. We conclude our review by briefly discussing some key aspects for the efficient implementation of the finite cell method.
An essential prerequisite for the efficient biomechanical tailoring of crops is to accurately relate mechanical behavior to compositional and morphological properties across different length scales. ...In this article, we develop a multiscale approach to predict macroscale stiffness and strength properties of crop stem materials from their hierarchical microstructure. We first discuss the experimental multiscale characterization based on microimaging (micro-CT, light microscopy, transmission electron microscopy) and chemical analysis, with a particular focus on oat stems. We then derive in detail a general micromechanics-based model of macroscale stiffness and strength. We specify our model for oats and validate it against a series of bending experiments that we conducted with oat stem samples. In the context of biomechanical tailoring, we demonstrate that our model can predict the effects of genetic modifications of microscale composition and morphology on macroscale mechanical properties of thale cress that is available in the literature.
•A micromechanics model for stiffness and strength of hierarchical culm materials is proposed.•Each hierarchical level is characterized by suitable microimaging data.•The model predictions are in ...excellent agreement with experimental measurements.•The model enables a physics-based understanding of the origins of bamboo properties.
Plant materials exhibit a wide range of highly anisotropic mechanical behavior due to a hierarchy of microheterogeneous structures at different length scales. In this article, we present a micromechanics approach that derives a hierarchical microstructure driven model of macroscopic stiffness and strength properties of anisotropic culm materials. As model input, it requires mechanical properties of the base constituents such as cellulose and lignin as well as morphology and volume fractions of all heterogeneous components at each hierarchical level. The latter can be retrieved from imaging data at different length scales, obtained from scanning electron and transmission electron microscopy. We illustrate our modeling approach for the example of bamboo that has gained increasing attention in the last decade due to its role as a sustainable building material. Validating its predictions of macroscopic stiffness moduli and ultimate strength with corresponding experimental measurements, we demonstrate that the micromechanics model provides excellent accuracy without any further phenomenological calibration. We also show that the multiscale modeling approach enables a better physics-based understanding of the origins of bamboo stiffness and strength across different scales.
The concept of concurrent material and structure optimization aims at alleviating the computational discovery of optimum microstructure configurations in multiphase hierarchical systems, whose ...macroscale behavior is governed by their microstructure composition that can evolve over multiple length scales from a few micrometers to centimeters. It is based on the split of the multiscale optimization problem into two nested sub-problems, one at the macroscale (structure) and the other at the microscales (material). In this paper, we establish a novel formulation of concurrent material and structure optimization for multiphase hierarchical systems with elastoplastic constituents at the material scales. Exploiting the thermomechanical foundations of elastoplasticity, we reformulate the material optimization problem based on the maximum plastic dissipation principle such that it assumes the format of an elastoplastic constitutive law and can be efficiently solved via modified return mapping algorithms. We integrate continuum micromechanics based estimates of the stiffness and the yield criterion into the formulation, which opens the door to a computationally feasible treatment of the material optimization problem. To demonstrate the accuracy and robustness of our framework, we define new benchmark tests with several material scales that, for the first time, become computationally feasible. We argue that our formulation naturally extends to multiscale optimization under further path-dependent effects such as viscoplasticity or multiscale fracture and damage.
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