^{1}

^{*}

^{1}

^{2}

^{1}

In this paper we present numerical simulations of soil plasticity using isogeometric analysis comparing the results to the solutions from conventional finite element method. Isogeometric analysis is a numerical method that uses nonuniform rational B-splines (NURBS) as basis functions instead of the Lagrangian polynomials often used in the finite element method. These functions have a higher-order of continuity, making it possible to represent complex geometries exactly. After a brief outline of the theory behind the isogeometric concept, we give a presentation of the constitutive equations, used to simulate the soil behavior in this work. The paper concludes with numerical examples in two- and three-dimensions, which assess the accuracy of isogeometric analysis for simulations of soil behavior. The numerical examples presented show, that for drained soils, the results from isogeometric analysis are overall in good agreement with the conventional finite element method in two- and three-dimensions. Thus isogeometric analysis is a good alternative to conventional finite element analysis for simulations of soil behavior.

In the design of foundations and geotechnical structures it is essential to predict soil behavior under different loading conditions. In the last decade finite element analysis (FEA) has become a widely spread tool for predicting soil behavior. Much research has been carried out to improve the ability of simulating the behavior of different soils and a number of new constitutive models have been developed. Another important aspect when modeling geotechnical problems is the interaction between soil and structure, which can have a large influence on the structural design [

Since first introduced by Hughes et al. in [_{0} finite element analysis with regard to computational costs [

In this work we have evaluated how isogeometric analysis performs compared to the conventional finite element method for soil plasticity in two and three dimensions. The focus of the work is to evaluate the convergence behavior and mesh size dependencies for isogeometric analysis and to compare them with results from conventional FEA. To provide a background, we start with outlining the basic concepts of isogeometric analysis, introducing the definitions of B- splines and NURBS that are used as basis or shape functions in this work. We later continue with the basic governing equations for elasto-plasticity followed by a brief recapitulation of the Drucker-Prager constitutive equations. The paper is concluded with two- and three-dimensional numerical examples, comparing the results to conventional finite element analysis.

The fundamental idea behind IGA is to employ the same basis functions for both geometrical discretization and analysis. Herein lays also the most profound difference between IGA and standard FEA, where isogeometric analysis utilizes the basis functions from CAGD, capable of representing the exact geometry also for analysis. Whereas, in conventional finite element analysis, the piecewise polynomials chosen to approximate the solution fields are also used to approximate the geometry [

To get an overview of the concept of NURBS-based isogeometric framework used in this work we start by defining a B-spline curve. A B-spline curve is a linear combination of B-spline basis functions,

The B-spline basis functions N_{i,p} are constructed from a non-decreasing set of coordinates in the parameter space, written as

For

which is referred to as the Cox-de Boor formula, first presented in [_{i,p}(ξ) ≥ 0. Another important aspect of spline basis functions, that distinguishes them from conventional FEA basis functions, is that each p^{th} order function has p − 1 continuous derivatives across element boundaries.

One issue that arises when using B-spline, is that not all types of geometries can be represented exactly using polynomial functions. To overcome this problem rational B-splines were introduced by Versprille in [_{i}, to each control point and making use of the definition of rational functions as the ratio of two polynomials [_{i}, are the d + 1 components of projective control points. Non-Uniform Rational B-splines (NURBS) are today standard in many CAGD software’s and the fast and stable algorithms make them a good choice also for analysis. To construct NURBS basis functions one can make use of the basis functions for B-splines,

where W(ξ) is called a weighting function, defined as

If w_{i} = 1 for all i, then _{i} = a for all i then _{i}, a piecewise NURBS curve, is constructed from

To be able to perform two- and three-dimensional analysis, NURBS surfaces and bodies needs to be defined. This is done in a similar manner. A NURBS body is given by

where the NURBS basis function,

where

In the isogeometric framework the concept of elements is represented by the non-zero valued knot spans. This is illustrated in _{i,j} and their corresponding weights w_{i,j}. The geometry is constructed of one knot vector, Ξ = {0, 0, 0, 0.5, 1, 1, 1} in the direction of the arc of the hole and one knot vector,

The number of active functions in a knot span is determined as

In this section we give a brief recapitulation of the strong and weak form of the equilibrium equations of the quasi-static balance of linear momentum. Let u_{i} denote the displacement vector, then the infinitesimal strain tensor, ε_{ij}, is defined as the symmetric part of the displacement gradient

The governing strong form of the equilibrium equation is given as

where f_{i} is the body force. The strong form of the equilibrium equation is complemented by the essential and natural boundary conditions.

where g_{i} and h_{i} are known quantities on ∂Ω_{g} and ∂Ω_{h}. The variational or weak form of Equation (11) is formed by multiplying the governing equation with a test function v_{i} and performing an integration by parts over the domain Ω

The weak form of the problem is complemented by a constitutive relation

with

The main difference between conventional FEA and IGA are the basis functions used for discretization. In isogeometric analysis the same basis functions that are used to discretize the geometry are also used to solve for the approximate displacement field u. The only difference to conventional FEA is that the basis functions in IGA are element-specific. After solving the element-specific basis functions and their derivatives the procedure of establishing the stiffness matrix and internal force vector is identical to conventional FEA. The element specific basis function for each element is determined as the non-zero functions in the knot span. The displacement field for any given element can be solved using the element specific basis function

with

To obtain the derivatives of the basis functions with respect to the physical coordinates one must use the chain rule,

with^{e} contains the basis functions

In a similar manner we rewrite Equation (17) in matrix form,

where the matrix B^{e} is an operator mapping the element discrete displacements to the local strains.

The test function v_{i} can be discretized in the same manner as the displacement field. Introducing the relations above it is possible to rewrite Equation (14) in a residual format as

with a the global unknown displacement vector. In this work we use a Newton-Raphson method to solve this system of nonlinear equations. The Jacobian matrix for the Newton-Raphson method reads

with D_{ats} the Voigt representation of the algorithmic tangent stiffness. In each Newton iteration we compute a displacement increment ∆a by solving the linear equation^{−6}.

In this section we will give a brief recapitulation of the Drucker-Prager criterion and the theory of plasticity used to evaluate the influence of the NURBS-basis functions on the plastic strains for granular materials in this work.

The Drucker-Prager criterion can be seen as a smooth approximation to the Mohr-Coulomb law and states that plastic yielding begins when the J_{2} and I_{1} invariants reach a critical combination. Although the Drucker-Prager formulation is a rather crude approximation of real soil behavior it has the benefit of being straightforward to implement and also lacks the singularities that exist in the yield function of the Mohr-Coulomb criterion. The yield function of the Drucker-Prager can be expressed using the first invariant of the stress tensor, I_{1}, and the second invariant of the deviatoric stress, J_{2},

which forms a circular cone in the principal stress space. The material parameters k and , can be expressed in terms of the materials cohesion,

where θ in represents the Lode angle. A positive sign in Equation (25) matches the Drucker-Prager cone to the inner edges of the Mohr-Coulomb surface. The matching to the Mohr-Coulomb criterion is illustrated in

We assume a small strain setting and thus additively split the strain tensor

For geo-materials in general, associative flow rules for the plastic strain

associative flow rule, that is, the potential function

The potential function is then written as

At the surface of the Drucker-Prager yield function the evolution of the plastic strain is determined as,

with

with S_{ij} the deviatoric part of the stress tensor. Furthermore we assume a bi-li- near hardening model for the material cohesion

The presented model is complemented by the loading/unloading conditions

for our plasticity model as_{ats}. We like to mention that at the apex of the yield surface cone, the return vector is contained by a complementary cone, illustrated in

To validate the performance of isogeometric analysis for soil-plasticity, three numerical benchmark models have been established. The models have all been simulated for soils in saturated conditions using the Drucker-Prager criterion. The models evaluated consist of one two-dimensional model of a strip footing and two three-dimensional cylindrical soil profiles subjected to a prescribed force and prescribed displacement respectively.

A two-dimensional model of a strip footing on sandy silt has been analyzed. Due to the symmetric properties of the problem the analysis contains only half of the footing. The geometry and boundary conditions are shown in

The problem is solved for plane strain conditions using quadratic NURBS- based IGA and conventional FEA with 5 different meshes. In order to compare the two methods the element meshes have been constructed using quadratic isoparametric elements for both IGA and conventional FEA. The element size ranges from a coarse mesh with elements of 2 × 2 meters down to a mesh with elements of 0.25 × 0.25 meters. The mesh data and degrees of freedom for each mesh are shown in

min. el. size [m] | nel | ndofs (FEA) | ndofs (IGA) |
---|---|---|---|

2 × 2 | 75 | 682 | 238 |

1 × 1 | 300 | 2562 | 768 |

0.5 × 0.5 | 1200 | 9922 | 2728 |

0.33 × 0.33 | 2700 | 22082 | 5888 |

0.25 × 0.25 | 4800 | 39042 | 10248 |

Material Parameter | Value | [Unit] |
---|---|---|

Young’s modulus, E | 100 | [Mpa] |

Poisson’s ratio, ν | 0.3 | - |

Cohesio, _{ } | 20 | [kPa] |

Cohesio, _{ } | 23 | [kPa] |

Angle of internal friction, _{ } | 20 | [deg] |

Angle of internal friction, _{ } | 22 | [deg] |

angle of dilatation, ψ | 5 | [deg] |

Accumulated plastic strai, _{ } | 0.04 | - |

that there is a minor difference between the displacements from the isogeometric- and conventional finite element analysis at the edge of the footing (x = 2).

The effects of the continuous stress fields that result from the NURBS basis functions can be seen in

results from the finest mesh used. The graphs show that the convergence for the isogeometric analysis and the conventional finite element analysis are comparable.

The two three-dimensional studies in this work are composed of a cylindrical soil profile with a 2:1 height/diameter proportion. In the first example, the cylindrical soil profile is subjected to a prescribed displacement at the top of the cylinder, in the second study the soil profile is subjected to a confining pressure and an increased vertical load at the top of the cylinder. The geometry and boundary conditions of the three-dimensional examples are presented in

Material Parameter | Value | [Unit] |
---|---|---|

Young’s modulus, E | 1000 | [kpa] |

Poisson’s ratio, ν | 0.3 | - |

Cohesio, c’ | 5.5 | [kPa] |

Angle of internal friction, ϕ’ | 10 | [deg] |

angle of dilatation, ψ | 5 | [deg] |

In the displacement controlled analysis a total strain of 7% is applied over 100 equal time by prescribing a displacement at the top surface. To compare the results from the isogeometric analysis with the finite element analysis,

To study the effects of the mesh density,

stress components over the top surface of the soil profile for each mesh normalized to the vertical stress from the finest mesh. From

In the force-controlled example the soil profile is subjected to a confining pressure, σ_{c}, and an axial load, σ_{v}, acting on the top surface as illustrated in _{c} will be more accurate using a soil profile made out of a perfect cylinder, than one discretized with Lagrangian polynomials. This can be seen in

The aim of this work is to evaluate the isogeometric framework for numerical

analysis of soil behavior. To compare IGA to conventional FEA, the Drucker- Prager criterion has been implemented together with a NURBS-based isogeometric framework. A numerical study has been conducted using NURBS-based isogeometric analysis comparing the results with results from analysis performed using the finite element method. The numerical examples presented show, that for drained soils; the results from isogeometric analysis are overall in good agreement with the conventional finite element method in two- and three-dimensions. However, the results from the two-dimensional example presented illustrate that the higher continuity of the basis functions used in IGA can have an effect on the plastic strains where abrupt stress changes can be expected. For the three- dimensional examples presented in this paper the isogeometric analysis has performed as good as or better than the finite element method, comparing the load/displacement response. To compare the computational efficiency of IGA and conventional FEA is not within the scope of this study. Further, geotechnical applications like retaining walls in weak soil or installations of friction piles often involve complicated contact problems and fluid flow, hence could benefit from using the isogeometric framework.

The Development Fund of the Swedish Construction Industry, SBUF, supported this work. The support is gratefully acknowledged.

Spetz, A., Tudisco, E., Denzer, R. and Dahlblom, O. (2017) Isogeometric Analysis of Soil Plasticity. Geomaterials, 7, 96-116. https://doi.org/10.4236/gm.2017.73008