Lagrangian Discontinuous Galerkin Hydrodynamic Method Research Articles

Simulation of dynamic crystal plasticity with a Lagrangian discontinuous Galerkin hydrodynamic method

We present a new Lagrangian modal discontinuous Galerkin (DG) hydrodynamic method that supports a dynamic dislocation based crystal plasticity model for simulating the mechanical behavior of crystallographic materials, both single crystal and polycrystalline, under dynamic conditions. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated by Taylor series polynomials of varying degree. These polynomials describe macro-scale hydrodynamic behavior while their evolution is determined by evaluating the dynamic crystal plasticity model at material points within the element. The dynamic crystal plasticity model is sensitive to the time increment size, with too large of time increments leading to instability in the model. To mitigate this, the temporal evolution of the dynamic crystal plasticity model is achieved with the combination of a sub-incrementing scheme with Heun’s third-order time integration scheme, which is also used to temporally evolve the governing equations. The implementation of the dynamic crystal plasticity model within the DG framework is tested using a 2D approximation of the Taylor impact test with a single crystal material, using quadratic elements that have faces that can bend. In addition to the standard continuous material modeling, we propose a new simulation method that would represent the heterogeneous behavior of polycrystalline microstructures within an element by varying the position and material properties of the material points within the element. This method is demonstrated using random orientation distributions on materials points that are arranged in both structured and random configurations.

A multiphase Lagrangian discontinuous Galerkin hydrodynamic method for high-explosive detonation physics

We present a new multidimensional multiphase Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports programmed burn and reactive burn high-explosive (HE) detonation models. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated by Taylor series polynomials of varying degree. The HE reaction mass fraction is represented using a nodal quantity. A new limiting approach is presented along with new special limiting conditions for the reactive burn model. The HE models are applied to individual quadrature points in the DG elements to better resolve the physics with the modal method. In the programmed burn model, this means that reaction times are calculated for each point separately. The Scaled Unified Reactive Front (SURF) reactive burn model uses a lead shock pressure to determine the reaction rate at the quadrature points, with a new lead shock pressure detector that works with the Taylor series polynomials. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. The robustness of the new limiting approach is tested using 1D and 2D strong shock gamma-law gas problems. The accuracy of the point-wise programmed burn model is tested using a 1D gamma-law gas verification problem and a 2D radial problem. The implementation of the SURF reactive burn model within the DG method is tested using a 1D shock-to-detonation problem simulating the behavior of PBX-9501 and a 2D confined slab problem simulating the behavior of PBX-9502, both represented by the Davis equations of state with a multiphase modeling method. All 2D results use quadratic elements, which have faces that can bend.

A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for solid dynamics

We present a new multidimensional high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher-order elements. We also present new one-dimensional test problems that have an analytic solution corresponding to a hyperelastic–plastic wave. A modal DG approach is used to evolve fields relevant to conservation laws. These fields are approximated high-order Taylor series polynomials. The stress fields are represented using nodal quantities. The constitutive models used to calculate the deviatoric stress are either a hypoelastic–plastic, infinitesimal strain hyperelastic–plastic, or finite strain hyperelastic–plastic model. These constitutive models require new methods for calculating high-order polynomials for the velocity gradient and deformation gradient in an element. The plasticity associated with the strength model is determined using a radial return method with a J2 yield criterion and perfect plasticity. The temporal evolution of the governing equations is achieved with the total variation diminishing Runge–Kutta (TVD RK) time integration method. A diverse suite of 1D and 2D test problems are calculated. The new 1D piston test problems, which have analytic solutions for each elastic–plastic model, are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new Lagrangian DG method. 2D test problems are calculated to demonstrate the stability and robustness of the new Lagrangian DG method on multidimensional problems with high-order elements, which have faces that can bend.

A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme

We present a Lagrangian discontinuous Galerkin (DG) hydrodynamic method that is up to third-order accurate using subcell mesh stabilization (SMS) for compressible flows on quadratic meshes in two-dimensional (2D) Cartesian coordinates. Similar to the second-order accurate Lagrangian DG method with linear meshes, the physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with Taylor series polynomials. The Riemann velocity at the vertices of a curvilinear cell, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem. Curvilinear cells (e.g., quadratic quadrilateral meshes in this work) have many deformational degrees of freedom, and with these cells, they can deform in unphysical ways. Likewise, the Riemann solution at an edge vertex differs from the one at the corner of a cell. With SMS, each quadratic quadrilateral cell is decomposed into four quadrilateral subcells, that move in a Lagrangian manner. The edge vertex is surrounded by four subcells so that it is similar to the vertex at the cell corner. SMS can detect inconsistent density fields between the cell and subcells. The difference between these two density fields is used to correct the stress (pressure) input to the Riemann solver. This SMS scheme enables stable mesh motion and accurate solutions in the context of a Lagrangian high-order DG method that is up to third-order with quadratic cells. We also present effective limiting strategies that ensure monotonicity of the primitive variables with the high-order DG method. This Lagrangian DG hydrodynamic method with SMS conserves mass, momentum, and total energy. A suite of test problems are calculated to demonstrate the designed order of accuracy (up to third-order accurate) of this method, and that the Lagrangian DG method using SMS preserves cylindrical symmetry on 1D radial flow problems with an equal-angle polar quadratic mesh.

A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for elastic–plastic flows

We present a new high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas and solid dynamics. The evolution equations for specific volume, momentum, and total energy are discretized using the modal DG approach. The specific volume, velocity, and specific total energy fields are approximated with up to quadratic Taylor series polynomials. The specific internal energy, pressure, and stress deviators are nodal quantities. The stress deviators are evolved forward in time using a hypoelastic–plastic approach, which requires a velocity gradient. A new method is presented for calculating a high-order polynomial for the velocity gradient in an element. Plasticity is handled by applying a radial return model to the stress deviators. Limiting approaches are presented for modal and nodal fields. The TVD RK time integration method is used to temporally advance all governing evolution equations. Generalized Lagrangian DG equations are derived but test problems are calculated for 1D Cartesian coordinates. A suite of gas and solid dynamics test problem results are calculated to demonstrate the stability and formal accuracy of the new Lagrangian DG method.

An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics

We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ,ρu,E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energy (ρ,u,e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters.

A Lagrangian discontinuous Galerkin hydrodynamic method

We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The third approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.