Meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility. Especially the tremendous progress in processor technology within recent years ...relativizes the increase in computation time due to the inherent search algorithm. Nevertheless meshfree approximation schemes are still faced with some challenges, like imposition of Dirichlet boundary conditions, robustness of the algorithm and accuracy. The recent developed Optimal Transportation Meshfree (OTM) method seemed to overcome most of these problems. In this paper the OTM solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation. However this scheme is not stable for some examples of application. Hence an investigation is conducted which shows that the reason for this instability is due to underintegration. Based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the Finite Element Method. In contrast to the original publication, the OTM algorithm is derived here from the principle of virtual work. Local maximum entropy shape functions are used which possess a weak Kronecker-δ property. This enables a direct imposition of Dirichlet boundary conditions if the boundary is convex. The limitations of this basis function are also addressed in this paper. Additionally, the search algorithm presented fulfills basic topological requirements. Several examples are investigated demonstrating the improved behavior of the stabilized OTM algorithm.
The Peridynamic Petrov–Galerkin (PPG) method is a meshfree approach based on the peridynamic integro-differential form of the momentum equation. The spurious oscillations in the common peridynamic ...correspondence formulation are investigated. They occur due to an inadmissible linearized mapping of the family deformation field. This leads to a generalized correspondence formulation, which contains the common formulation as a special case. It is based on the weak form of the peridynamic momentum equation. Test and trial function requirements are examined which ensure an exact imposition of Dirichlet and Neumann boundary conditions and Weighted Least Square (WLS) shape functions as well as Local Maximum Entropy (LME) approximants are utilized to examine the PPG Method. A consistent linearization is provided, which can also be used to speed up common implicit peridynamic correspondence codes. It is used in an implicit quasistatic framework to investigate the impact of different shape function combinations. Test cases show that low-energy modes can be prevented by the PPG Method and highlight the fast convergence and stability.
•An ansatz function based correspondence formulation eliminates low-energy modes.•Ansatz function conditions enforce accurate boundary conditions and convergence.•Consistent linearization enables efficient implicit peridynamic correspondence codes.
Selective Laser Melting (SLM) is an Additive Manufacturing (AM) process where a powder bed is locally melted. Layer by layer, complex three dimensional geometries including overhangs can be produced. ...Non-melted powder thereby acts as support structure. The process is held under an inert gas atmosphere to prevent oxidation. The principal machine parameters in SLM processes are the laser power, the scan rate and the laser spot radius. The powder bed is characterized by the material, the packing density and the particle size distribution. These factors define the structure of SLM finished parts. Up to date, the material and process development of SLM mainly relies on experimental studies that are time intensive and costly. Simulation tools offer the potential to gain a deeper understanding of the process–structure–property interaction. This can help to find optimal process parameters and to individualize AM manufactured parts.
A continuum framework for the finite deformation phase change problem is developed. For its numerical solution the stabilized Optimal Transportation Meshfree Method (OTM) is employed. The advantage of meshfree over conventional mesh based techniques is that the treatment of particle fusion is intrinsic to the formulation. This is important to resolve the complex moving boundaries between liquid melt flow and solid metal. In a numerical example consisting of two metal powder particles, the influence of laser heating and cooling conditions on melting and consolidation is analyzed. A detailed parameter study is presented. The insight gained from the simulations may help to narrow the parameter window for further investigations.
For the accurate imposition of surface loads using the Finite Element Method normally the load is discretized at the surface facets. Therefore appropriate surface shape functions are needed. If the ...surface contains kinks, which occurs often in contact cases, the imposition is more complicated. In order to simplify the imposition of surface loads, an alternative approach is presented. This formulation is purely based on volume contributions of the discretized elements of the body. The surface nodes are automatically identified and the linearization is straightforward. No specific surface information is necessary. The imposition is simply based on the application of the divergence theorem. The only prerequisite is the fulfillment of the integration constraint, a necessary requirement for Galerkin solution schemes. With this, boundary nodes are directly identified by possessing a non zero normal vector whereas for inner nodes this vector is identically zero. Moreover, the normal vectors at the boundary nodes contain all the information of the surface and correspond to resultant nodal normal vectors. This is especially advantageous in the case of surfaces that contain kinks. It also simplifies the search algorithm in computational contact mechanics to find the closest distance to other bodies. This approach also works for any kind of shape functions. The nodal force vectors are always determined accurately. The advantages and the simple handling of this approach are demonstrated by means of several examples including follower loads and contact cases. Additionally, the influence of the isoparametric concept on the integration constraint is investigated by evaluating the behavior of different shape functions on an irregular grid. Although this new approach is only demonstrated within the context of the Finite Element Method, due to its generic derivation it can be applied to any Galerkin solution scheme which fulfills the integration constraint.
•Automatic identification of boundary and inner nodes without any information about the surface.•Automatic computation of resultant nodal normal vectors at the surface even in case of surfaces containing kinks.•Simple computation of nodal force vectors within the FEM based on volume information only.
This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary ...particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker-
δ
property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.
The Peridynamic Petrov–Galerkin (PPG) method is a meshfree particle method based on the weak form of the peridynamic momentum equation. It can be applied to arbitrary constitutive laws from the ...classical continuum mechanics theory. With non-linear approximation functions the rank deficiency present in many nodally integrated discretization schemes is prevented. The consistency of trial functions is not sufficient for the convergence with irregular particle distributions. In this paper the consistency of the test space is examined and possible correction techniques are presented. The resulting variationally consistent PPG method is able to pass the patch test and to restore the optimal convergence rates. A correction of the test functions that preserves the linear trial function consistency allows the use of displacement–pressure–dilation formulations and exhibits stability and robustness for 3-D in the regime of non-linear elasticity. Besides, the direct nodal coupling with Finite Elements and the application of symmetry boundary conditions are enabled.
•Variationally consistent PPG formulations satisfying patch tests•Preservation of linear consistency yields compatibility with mixed approaches•Restored optimal convergence rate for irregular particle distributions•Direct coupling with Finite Elements and symmetry boundary conditions.
The large flexibility of meshfree solution schemes makes them attractive for many kinds of engineering applications, like Additive Manufacturing or cutting processes. While numerous meshfree methods ...were developed over the years, the accuracy and robustness are still challenging and critical issues. Stabilization techniques of various kinds are typically used to overcome these problems, but often require the tuning of unphysical parameters. The Peridynamic Petrov–Galerkin method is a generalization of the peridynamic theory of correspondence materials and offers a stable and robust alternative. In this work, the stabilization free approach is extended to three dimensional problems of finite elasticity. Locking-free mixed formulations for nearly incompressible and incompressible materials are developed and investigated in convergence studies. In general, an efficient implicit quasi-static framework based on Automatic Differentiation is presented. The numerical examples highlight the convergence properties and robustness of the proposed formulations.
Selective Laser Melting (SLM) is an emerging Additive Manufacturing technology for metals. Complex three dimensional parts can be generated from a powder bed by locally melting the desired portions ...layer by layer. The necessary heat is provided by a laser. The laser–matter interaction is a crucial physical phenomenon in the SLM process. Various modeling approaches with different degrees of complexity exist in the literature to represent the laser–matter interaction within a numerical framework. Often, the laser energy is simply distributed into a specified volume. A more precise approach is ray tracing. The laser beam can be divided into moving discrete energy portions (rays) that are traced in space and time. In order to compute the reflection and absorption usually a triangulation of the free surface is conducted. Within meshfree methods, this is a very expensive operation. In this work, a computationally efficient algorithm is developed which avoids triangulation and can easily be combined with meshfree methods. Here, the suggested ray tracing algorithm is exemplary coupled with the stabilized Optimal Transportation Meshfree Method. The importance of ray tracing is evaluated by simulating the fusion of metal powder particles. A comparison of the results with a volumetric heat source approach shows that ray tracing significantly improves the accuracy of absorption and vaporization.
In many contact situations the material behavior of one contact member strongly influences the force acting between the two bodies. Unfortunately standard friction models cannot reproduce all of ...these material effects at the contact layer and often continuum interface elements are used instead. These elements are intrinsically tied to the fixed grid and hence cannot be used in large sliding simulations. Due to the shortcomings of the standard contact formulations and of the interface elements a new type of a contact layer element is developed in this work. The advantages of this element are the direct implementation of continuum models into the contact formulation and the application to arbitrary large deformations. Showing a relation between continuum and contact kinematics based on the solid-shell concept the new contact element is at the end a natural extension of the standard contact formulations into 3D. Two examples show that the continuum behavior can be exactly reproduced at the contact surface even in large sliding situations using this contact layer element. For the discretization of the new contact element the Mortar method is chosen exemplary, but it can be combined with all kinds of contact formulations.
In this work two new concepts for a direct application of plasticity models within a frictional contact description are developed. These concepts can be used in conjunction with all different kinds ...of contact formulations and solution methods. Additionally, all types of plasticity models can be projected onto the contact surface. The advantage of these concepts is shown exemplary in the modeling process of soil-structure interactions where the projected plasticity models are able to describe the soil behavior at the contact surface. The numerical implementation of the new frictional relations is based on the Mortar method. A new type of mixed formulation is also introduced combining the augmented Lagrangian method to enforce the normal contact constraint with the penalty regularization written in Hellinger–Reissner form to implement the tangential contact behavior. This reformulation leads to a reduction of the CPU time compared to the standard penalty regularization, if the Mortar method is used. Finally, the numerical investigation of a direct shear test shows the accurate reproduction of the typical stress–strain relation of the soil at the contact surface.
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