The Discontinuous Galerkin Material Point Method for variational hyperelastic–plastic solids

Renaud, A.; Heuzé, Thomas; Stainier, Laurent

doi:10.1016/j.cma.2020.112987

Cited by 8 publications

(5 citation statements)

References 46 publications

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“…No step of projection of internal variables from one grid to another is required as in classical mesh‐based methods. ADER‐DGMPM can reduce to the first‐order DGMPM if wished, which thus represents a lower limit (actually of first‐order accuracy) to the approximation of arbitrary high order of ADER‐DGMPM. This can be achieved by changing the order of the MLS approximation (actually using Shepard's functions (54)) and using an integration rule based on material points 38,40 rather than on Gauss–Legendre integration points. The main interest of the existence of such first‐order accurate lower limit lies in that it can provide a monotone approximation, which appears as another way of achieving a non‐oscillatory solution in the vicinity of shocks than using limiters.…”

Section: Discussionmentioning

confidence: 99%

“…The limited solution is reconstructed componentwise. It involves the reconstructed gradient (40), and a set of limiting coefficients associated with each component of system (5) gathered in the vector 𝜶 e , such that dim(𝜶 e ) = M, where dim(•) refers to the dimension of the quantity (•). The reconstructed limited solution reads:…”

Section: Basic Slope Limitersmentioning

confidence: 99%

“…One example is the smooth particle hydrodynamics method, 34,35 applied for solid mechanics problems with a conservative formulation in the isothermal setting, 36 then extended to thermoelasticity. 37 Another example is the discontinuous Galerkin Material Point Method (DGMPM), [38][39][40] which combines the DGFEM 41 with the material point method 42 (MPM). The latter comes from the family of particle-in-cell methods (PIC) 43 developed in the 1960s.…”

Section: Introductionmentioning

confidence: 99%

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ADER discontinuous Galerkin Material Point Method

Lakiss,

Heuzé,

Tannous

et al. 2023

Numerical Meth Engineering

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The first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

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Section: Discussionmentioning

confidence: 99%

Section: Basic Slope Limitersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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ADER discontinuous Galerkin Material Point Method

Lakiss,

Heuzé,

Tannous

et al. 2023

Numerical Meth Engineering

Self Cite

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“…Currently, this parameter is determined mainly by probabilistic methods. An attempt to determine the full (maximum) load-bearing capacity of structures using the Lagrange approach [8][9][10][11][12][13][14] leads to consideration of all possible options for loading load-bearing structures, i.e. the need to solve an infinite sequence of problems.…”

Section: Introductionmentioning

confidence: 99%

Determination of the Residual Strain Energy of a Structure and the Method of Progressive Limiting State

Stupishin,

Nikitin

2023

Advances in Transdisciplinary Engineering

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The technique of progressive limiting state of the structure, which is implemented in the form of the displacement method, is considered. The existing formulations of the problems in structural mechanics are based on the principle of the minimum of the total energy of deformation of structures (Lagrange approach), according to which the resulting project corresponds to a predetermined load. But using this approach, it was not possible to obtain information about its residual load-bearing capacity in a deterministic form. To overcome these difficulties, it is proposed to use the criterion of critical levels of deformation energy. The equations of state of the structure at critical energy levels are derived from the variational principle of the minimum of the potential energy of deformation at the critical energy level. The problem is formulated as an eigenvalue problem and the values of the maximum values of the design parameters are found for the stiffness matrix. It becomes possible to calculate the displacements for the critical energy level, and the maximum possible value of the deformation energy of the structure. Subtracting the magnitude of the work of external forces, we obtain the value of the residual potential energy, which determines the residual load-bearing capacity of the structure. When analyzing internal forces in a structure in a critical condition, we determine their greatest values, which cause the limiting state. We call these elements the ‘weak link’ because they will be the first to stop working on an external load. Sequentially removing these elements from the design scheme, we obtain new states of self-tension of the structure until it becomes an unstable system. Thus, we obtain the technique of progressive limiting state of the structure.

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“…Nowadays, almost all calculations associated with the estimation of ultimate states reached by a structure during designing of structures are performed in the Lagrangian form [1][2][3][4][5][6]. This formulation of the problem allows us to obtain the design of load-bearing structures of a construction only for specified values of loads.…”

Section: Introductionmentioning

confidence: 99%

Methodology for determining progressing ultimate states based on the displacement method

Stupishin,

Nikitin,

Moshkevich

2023

Struct. Mech. of Eng. Const. and Build

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Solving of calculation problems for building structures is currently based on the principle of minimum total energy of structures deformation. However, it is not possible to determine the remaining bearing capacity of the structure using this principle. In the study it is proposed to use the criterion of critical levels of deformation energy to solve this problem. As a result, the ultimate state conditions of a design are formulated on the basis of extreme values of generalized parameters of designing over the whole area of their admissible values, including the boundary. The task is solved as a problem of eigenvalues for the stiffness matrix of the system. The extreme values of design parameters that correspond to critical energy levels are found, which are used to find the maximum possible value of the energy of deformation for the considered structure. The residual bearing capacity is calculated by the value of residual potential energy, which, in turn, is equal to the difference between the maximum possible value of the deformation energy of the structure and the work of external forces. A gradual methodology for investigating the progressive ultimate limit state is proposed, which is based on the sequential exclusion of those elements where the onset of the ultimate limit state is expected firstly. An example of the practical use of the proposed methods is given on the example of calculating a simple but visual design - a statically indeterminate truss.

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The Discontinuous Galerkin Material Point Method for variational hyperelastic–plastic solids

Cited by 8 publications

References 46 publications

ADER discontinuous Galerkin Material Point Method

ADER discontinuous Galerkin Material Point Method

Determination of the Residual Strain Energy of a Structure and the Method of Progressive Limiting State

Methodology for determining progressing ultimate states based on the displacement method

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