Conference Agenda

Nonlinear multi-point constraints are used to model a wide range of features in engineering structures like incompressibility, inextensibility, rigidity or coupling of elements. The applied combination of such constraints can be either be non-redundant or redundant. There are three strategies to include nonlinear multi-point constraints in the finite element analysis: Lagrange multipliers, penalty method and master-slave elimination. Redundant nonlinear constraints are particularly challenging for all three methods. For Lagrange multipliers they can lead to a singular effective stiffness matrix and thus to divergence. For the penalty method redundancy worsens the convergence. For master-slave elimination redundant constraints lead to the selection of too many slave dofs or the wrong selection of master and slave dofs. This results in the failure of the method.

As the number of constraints in a given region of the system under study increases, the probability of the occurrence of redundant constraints also increases. For such problems it is not possible to detect the redundancy by hand because the coupling of constraints becomes very complex. An additional challenge is that the redundancy of constraints may change during the iterative solution process. Thus, redundant nonlinear constraints cannot be detected a priori in contrast to redundant linear constraints. Therefore, there is the need for an automatic detection and elimination of redundant nonlinear constraints during the simulation. However, in the literature redundancy has only been analyzed for certain combinations of constraints yet.

In this presentation we propose a direct, stable numerical method for the detection and elimination of arbitrary redundant nonlinear multi-point constraints that can be used for all three strategies. The corresponding treatment of the constraints has to be applied in each iteration. Thus, the presented method ensures the consistent linearization of the constraint forces and quadratic convergence is retained. We explain the algorithmic consequences on the three strategies due to the proposed method. In particular, we discuss the benefits of the method for the master-slave elimination as it gives also an automatic selection of master and slave dofs. We verify the method by selected examples and illustrate its influence on the numerical performance.

Isogeometric analysis (IGA) employs higher order polynomial shape functions, which are directly extracted from the CAD model. Unlike the standard Finite Element Method (FEM), which commonly uses Lagrange basis functions, IGA typically uses Non-Uniform Rational B-Splines (NURBS) or other types of splines. The use of spline-based FEM results in efficient computations with a relatively small number of elements, as increasing the order of NURBS basis functions enhances the convergence rate.

In the field of IGA, using high polynomial orders results in precise computations for structures that are exposed to static and dynamic loads. However, this also leads to highly accurate mass matrices with large bandwidths, resulting in increased computational effort, particularly for explicit dynamic analysis with a large number of time steps. To address computational efficiency, mass lumping techniques are commonly applied to achieve diagonal mass matrices, reducing the inversion of the mass matrix to a simple reciprocal operation. Many mass lumping schemes have been developed over time, but commonly the row-sum technique is attracted. These techniques were primarily invented fitting the requirements of standard FEM. Unfortunately, they deteriorate the convincing convergence rates of IGA when high polynomial orders are employed. Therefore, a lumping scheme tailored to IGA formulations using higher-order basis functions is necessary for efficient dynamic computations.

This study proposes the usage of dual basis functions as test functions in IGA element formulations with NURBS shape functions. By incorporating dual test functions, the Bubnov-Galerkin formulation is transferred to a Petrov-Galerkin formulation, resulting in non-symmetric stiffness matrices and - as effect of the duality - consistent diagonal mass matrices. For the presented formulation only dual basis functions are considered, which can be constructed by a combination of the initial NURBS shape functions. Hence, the changes by the dual formulation on element level can be shifted to an afterwards modification of the global matrices obtained from common IGA formulations. Thus, implementing this technique in existing codes is straightforward. Numerical examples demonstrate the efficiency of the dual approach compared to existing methods.

In modern applications of computer-aided design (CAD) for the analysis of shell structures, isogeometric analysis (IGA) is a powerful tool to incorporate both design and analysis. However, when it comes to multi-patch structures, C1-continuity across patches is not naturally fulfilled and computation of Kirchhoff-Love shells is not straightforward since well-defined second-order derivatives are necessary for the analysis. Furthermore, trimming is a major problem as the mathematical underlying of the CAD surface is not inherently suitable for standard IGA.

The approach presented in this talk deals with a Kirchhoff-Love shell formulation in the framework of scaled boundary isogeometric analysis [1,2] with C1-coupling. In scaled boundary (SB), the domain is described by its boundary and scaled to a scaling center, which we denote as an SB block. Thereby, each SB block consists of several IGA patches. This has the advantage of being applicable to multi-patch structures with various numbers of edges or boundaries. Besides, a possible trimming curve is easily incorporated into the boundary representation. The domain can be subdivided into several SB blocks to obtain star convexity. However, even for a single SB block, C1-continuity is not fulfilled across the IGA patches within the SB block. To show the feasibility of the coupling approach involving SB parametrizations, it heeds the concept of analysis-suitable G1 parametrizations [3] combined with special consideration of the basis functions in the scaling center. The method is especially powerful when it comes to complex geometries that cannot be described by a single IGA patch which is outlined in several examples.

[1] C. Arioli, A. Shamanskiy, S. Klinkel, and B. Simeon, “Scaled boundary parametrizations in isogeometric analysis”, Comput. Methods Appl. Mech. Eng., vol. 349, pp. 576–594, 2019.

[2] M. Chasapi and S. Klinkel, “A scaled boundary isogeometric formulation for the elasto-plastic analysis of solids in boundary representation”, Comput. Methods Appl. Mech. Eng., vol. 333, pp. 475–496, 2018.

[3] A. Collin, G. Sangalli, and T. Takacs, “Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces”, Comput. Aided Geom. Des., vol. 47, pp. 93–113, 2016.

Additive manufacturing enables the fabrication of geometrically complex structures, giving rise to a research focus on tailoring structures and material properties using numerical simulation. In lightweight engineering, continuous fiber composites are in great demand due to their superior strength-to-weight ratio. However, their anisotropic material properties pose difficulties for additive manufacturing processes: Planar 3D printing restricts fiber placement to a 2D plane, limiting the complexity of fabricated parts; current design methods for non-planar 3D printing (e.g., robotic arm) lack automated design methods with a performant integration of numerical simulation.

Another well-known problem of 3D-printed fiber composites is uncertainties in material parameters. Improvements in the micro-structure have been shown through consolidation or post-treatment (heat/pressure), reducing material variability. However, these methods are challenging to apply in non-planar 3D printing. Another approach is the experimental quantification of uncertainties in the material to incorporate them into the numerical simulation.

With the aim to contribute to design optimization for non-planar 3D printing of fiber composites, we develop a robust optimization methodology that considers firstly, the anisotropic nature of fiber composites and secondly, material uncertainties. The method is based on a ground structure discretization of the design space with 1D elements. It uses a heuristic approach for design optimization based on an optimality criteria for robustness aiming for equal strain energy distribution in the structure. The optimization methodology further utilizes the Certain Generalized Stresses Method (CGSM) for stochastic modeling to study the influence of the material uncertainties on the structural design.

Redundancy, and thus the degree of static indeterminacy, plays an important role in the design of structural systems. According to Linkwitz and Ströbel, the distribution of static indeterminacy in the system can be described by the redundancy matrix. The redundancy contribution of an element quantifies the internal constraint of the surrounding structure on this element. The sum of the redundancy contributions of all elements is equal to the degree of statical indeterminacy of the entire structure. The extension of Ströbel's notion for discrete truss systems to frames and continua can provide valuable insight into the load-bearing properties of a structure and has the potential to become an exciting new branch in the classical field of structural analysis.

Obviously, statical indeterminacy and its distribution in a structure have a decisive influence on the structural behavior. Therefore, the redundancy matrix can be a good measure to understand and evaluate structural behavior. It can also be used for robust design optimization and assessment of imperfection sensitivity during the assembly process.

The computation of the redundancy matrix generally requires a high effort due to the necessity of expensive matrix operations. A closed-form expression for the redundancy matrix can be derived via a factorization that is based on singular value decomposition. For moderately redundant systems it proves to be computationally very efficient. For small modifications of the structure, such as adding, removing, and swapping elements, generic algebraic formulations can be derived for efficiently updating the redundancy matrix.

While the redundancy matrix concept has only been applied to linear analysis, it can be extended to the nonlinear regime. In this case, the contribution of the nonlinear load transfer can be analyzed separately. Furthermore, the redundancy matrix allows for a simple extension of the notion of static indeterminacy to nonlinear analysis.

Simulations for predicting critical process variables in machining applications have been carried out for years. One important simulation-based analysis class in this context is the Finite Element Method (FEM). It is challenging to model the process with FEM as the metal is subjected to extensive deformation at high strain rates and temperatures. This large deformation is primarily irreversible and requires a plastic material model. The so-called orthogonal cutting process is a good abstraction of machining applications, where only a 2D representation is considered. It involves a tool cutting through a workpiece, forming chips. The shape of these chips is a crucial validation criterion for the accuracy of the simulation. One way to improve the representation of geometries in FEM simulations is to utilize Isogeometric Analysis (IGA), where the classical Lagrangian basis functions are replaced by the basis of Non-Uniform Rational B-Splines (NURBS). As these splines are commonly used for the representation of geometries in CAD models, IGA bridges the analysis with the initial design geometry.

The workpiece is modeled with a plastic material model, the Johnson-Cook hardening model, which includes a strain-rate dependency. Another crucial detail to model is the contact between the tool and the workpiece. In this work, we model the tool as a rigid B-Spline and employ a penalty contact formulation. Our focus is to investigate the influence of employing Isogeometric Analysis for the chip-forming process and the resulting chips. Furthermore, we compare the results to a classical FEM approach.