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In the search for more efficient and less environmentally harmful cooling technologies, the field of magnetocalorics is considered a promising alternative. To generate cooling spans, rotating permanent magnet assemblies are used to cyclically magnetize and demagnetize magnetocaloric materials, which change their temperature under the application of a magnetic field. In this work, an axial rotary permanent magnet assembly, aimed for commercialization, is computationally designed using topology and shape optimization. This is efficiently facilitated in an isogeometric analysis framework, where harmonic mortaring is applied to couple the rotating rotor-stator system of the multipatch model. Inner, outer and co-rotating assemblies are compared and optimized designs for different magnet masses are determined. These simulations are used to homogenize the magnetic flux density in the magnetocaloric material. The resulting torque is analyzed for different geometric parameters. Additionally, the influence of anisotropy in the active magnetic regenerators is studied in order to guide the magnetic flux. Different examples are analyzed and classified to find an optimal magnet assembly for magnetocaloric cooling.

Straight beams, rods and trusses are common elements in structural and mechanical engineering, but recent advances in additive manufacturing now also enable efficient freeform fabrication of curved, deformable beams and beam structures, such as microstructures, metamaterials and conformal lattices. To exploit this new design freedom for applications with nonlinear mechanical behavior, we introduce an isogeometric method for shape optimization of curved 3D beams and beam structures. The geometrically exact Cosserat rod theory is used to model nonlinear 3D beams subject to large deformations and rotations. The initial and current geometry are parameterized in terms of NURBS curves describing the beam centerline and an isogeometric collocation approach is used to discretize the strong form of the balance equations. Then, a nonlinear optimization problem is formulated in order to optimize the positions of the control points of the NURBS curve that describes the beam centerline, i.e., the geometry or shape of the beam. To solve the design problem using gradient-based algorithms, we introduce semi-analytical, inconsistent analytical and fully analytical approaches for calculation of design sensitivities. The methods are numerically validated and their performance is investigated, before the applicability and versatility of our 3D beam shape optimization method is illustrated in various numerical applications, including optimization of an auxetic 3D metamaterial.

A geometrically nonlinear, shear-deformable 3D beam formulation with inelastic material behavior and its numerical discretization by a mixed isogeometric collocation method are presented. In particular, the constitutive model captures elasto-visco-plasticity with damage/softening from Mullin's effect, which applies to the modeling of metallic and polymeric materials, e.g., in additive manufacturing applications and meta-materials. The inelastic material behavior is formulated in terms of thermodynamically consistent internal variables for viscoelastic and plastic strains and isotropic and kinematic hardening variables, as well as accompanying evolution equations. A mixed isogeometric collocation method is applied for the discretization of the strong form of the quasi-static nonlinear differential equations. Thus, the displacements of the centerline curve, the cross-section orientations, and the stress resultants (forces and moments) are discretized as B-spline or NURBS curves. The internal variables are defined only locally at the collocation points, and an implicit return-mapping algorithm is employed for their time discretization. The method is verified in comparison to 1D examples as well as reference results for 3D beams. Furthermore, its applicability to the simulation of beam lattice structures subject to large deformations and instabilities is demonstrated.

Lattice structures are frequently found in nature and engineering due to their myriad attractive properties, with applications ranging from molecular to architectural scales. Lattices have also become a key concept in additive manufacturing, which enables precise fabrication of complex lattices that would not be possible otherwise. While design and simulation tools for stiff lattices are common, here we present a digital design and nonlinear simulation approach for additive manufacturing of soft lattices structures subject to large deformations and instabilities, for which applications in soft robotics, healthcare, personal protection, energy absorption, fashion and design are rapidly emerging. Our framework enables design of soft lattices with curved members conforming to freeform geometries, and with variable, gradually changing member thickness and material, allowing the local control of stiffness. We model the lattice members as 3D curved rods and using a spline-based isogeometric method that allows the efficient simulation of nonlinear, large deformation behavior of these structures directly from the CAD geometries. Furthermore, we enhance the formulation with a new joint stiffening approach, which is based on parameters derived from the actual node geometries. Simulation results are verified against experiments with soft lattices realized by PolyJet multi-material polymer 3D printing, highlighting the potential for simulation-driven, digital design and application of non-uniform and curved soft lattice structures.

This article presents a numerical framework to predict the mechanical behavior of knitted fabrics from their discrete structure at the fabric yarn level, i.e., the mesostructure, utilizing the hierarchical multiscale method. Due to the regular distribution of yarn loops in a knitted structure, the homogenization theory for periodic materials can be employed. Thus, instead of considering the whole fabric sample under loading, a significantly less computationally demanding analysis can be done on a repeated unit cell (RUC). This RUC is created based on simple structural parameters of knitted yarn loops and its fabric yarns are assumed to behave transversely isotropic. Nonlinear finite element analyses are performed to determine the stress fields in the RUC under tensile and shear loading. During this analysis, contact friction among yarns is considered as well as the periodic boundary conditions are employed. The macroscopic stresses then can be derived from the stress fields in the RUC by means of the numerical homogenization scheme. The physical fidelity of the proposed framework is shown by the good agreement between the predicted mechanical properties of knitted fabrics and corresponding experimental data.

Photopolymerization-based additive manufacturing methods like stereolithography and digital light processing only allow typically the monolithic fabrication of structures made from a single material. To overcome this limitation, grayscale digital light processing has been proposed for 3D printing of functionally graded materials. Here, this concept is extended to grayscale masked stereolithography (MSLA) printing using a LED light source and a LCD photomask to control the degree of photopolymerization of a UV-curable resin by varying grayscale pixels and thus light intensities. In this scale, tailorable hyperelastic material properties and functionally graded structures for finite deformations are realized. In this paper, the dependency of the resulting material properties on the parameters of the grayscale MSLA process is investigated and a grayscale-dependent hyperelastic material model is formulated. This parametric hyperelastic material model is fitted to the experiments and validated against experimental results for uniaxial tension and uniaxial compression tests. Then, functionally graded structures with tailored mechanical properties at finite deformations are designed and fabricated using grayscale MSLA printing. The hyperelastic material model is validated with experimental results for different geometries, showing good agreement between experimental tests and numerical calculations.

Generalized continuum mechanical theories such as second gradient elasticity can consider size and localization effects, which motivates their use for multiscale modeling of periodic lattice structures and metamaterials. For this purpose, a numerical homogenization method for computing the effective second gradient constitutive models of cubic lattice metamaterials in the infinitesimal deformation regime is introduced here. Based on the modeling of lattice unit cells as shear-deformable 3D beam structures, the relationship between effective macroscopic strain and stress measures and the microscopic boundary deformations and rotations is derived. From this Hill–Mandel condition, admissible kinematic boundary conditions for the homogenization are concluded. The method is numerically verified and applied to various lattice unit cell types, where the influence of cell type, cell size and aspect ratio on the effective constitutive parameters of the linear elastic second gradient model is investigated and discussed. To facilitate their use in multiscale simulations with second gradient linear elasticity, these effective constitutive coefficients are parameterized in terms of the aspect ratio of the lattices structures.

We present a novel method for the mechanical simulation of slender, elastic, spatial rods and rod structures subject to large deformation and rotation. We develop an isogeometric collocation method for the geometrically exact, nonlinear Cosserat rod theory. The rod centerlines are represented as spatial NURBS curves and cross-section orientations are parameterized in terms of unit quaternions as 4-dimensional NURBS curves. Within the isogeometric framework, the strong forms of the equilibrium equations of forces and moments of the discretized Cosserat model are collocated, leading to an efficient method for higher-order discretizations. For rod structures consisting of multiple, connected rods we introduce a formulation with rigid, quasi- G 1 -coupling. It is based on the strong enforcement of continuity of displacement and change of cross-section orientation at interfaces. We also develop a mixed isogeometric formulation, which is based on an independent discretization of internal forces and moments and alleviates shear locking for thin rods. The novel rod simulation methods are verified by numerical convergence studies. Further computational examples include realistic applications with large deformations and rotations, as well as a large-scale rod structure with several hundreds of coupled rods and complex buckling behavior.

Soft lattice structures and beam-metamaterials made of hyperelastic, rubbery materials undergo large elastic deformations and exhibit structural instabilities in the form of micro-buckling of struts under both compression and tension. In this work, the large-deformation nonlinear elastic behaviour of beam-lattice metamaterials is investigated by micromechanical nonlinear buckling analysis. The micromechanical 3D beam finite element model uses a primary linear buckling analysis to incorporate the effect of geometric imperfections into a subsequent nonlinear post-buckling analysis. The micromechanical computational model is validated against tensile and compressive experiments on a 3D-printed sample lattice structure manufactured via multi-material jetting. For the development and calibration of macroscale continuum constitutive models for nonlinear elastic deformation of soft lattice structures at finite strains, virtual characterization tests are conducted to quantify the effective nonlinear response of representative unit cells under periodic boundary conditions. These standard tests, commonly used for hyperelastic material characterization, include uniaxial, biaxial, planar and volumetric tension and compression, as well as simple shear. It is observed that besides the well-known stretch- and bending-dominated behaviour of cellular structures, some lattice types are dominated by buckling and post-buckling response. For multiscale simulation based on nonlinear homogenization, the uniaxial standard test results are used to derive parametric hyperelastic constitutive relations for the effective constitutive behaviour of representative unit cells in terms of lattice aspect ratio. Finally, a comparative study for compressive deformation of a sample sandwich lattice structure simulated by both full-scale beam and continuum finite element models shows the feasibility and computational efficiency of the effective continuum model.

We present a nonlinear multiscale modeling and simulation framework for the mechanical design of machine-knitted textiles with functionally graded microstructures. The framework operates on the mesoscale (stitch level), where yarns intermesh into stitch patterns, and the macroscale (fabric level), where these repetitive stitch patterns are composed into a fabric. On the mesoscale, representative unit cells consisting of single interlocked yarn loops, modeled as geometrically exact, nonlinear elastic 3D beams, are homogenized to compute their effective mechanical properties. From this data, a B-Spline response surface model is generated to represent the nonlinear orthotropic constitutive behavior on the macroscale, where the fabric is modeled by a nonlinear Kirchhoff–Love shell formulation and discretized using isogeometric finite elements. These functionally graded textiles with locally varying properties can be designed and analyzed by parameterizing the stitch value, i.e., the loop length of a single jersey stitch, and the knitting direction as mesoscopic design variables of the macroscopic response surface constitutive model. To validate the multiscale simulation and design approach, numerical results are compared against physical experiments of different tensile loading cases for various grading scenarios. Furthermore, the versatility of the method for the design of functionally graded textiles is demonstrated.