We present two multifield and one single-field variational principles for the initial boundary value problem of diffusion. Chemical potential and concentration appear as conjugate variables in the multifield formulations. The main importance of the proposed formulations is the approach used to generate the variational principles, where the framework of Generalized Standard Materials is used for constitutive laws while natural boundary conditions and the balance of mass are used as constraints of the optimization problem. This approach allows to derive such principles for multiphysic problems in a generic manner. A detailed derivation and analysis of the formulations are presented, where it can be seen their equivalence with the most common strong and weak forms of the problem using Fick’s laws along with the logarithmic mass action law. From the stationarity condition with respect to the mass flux of the initially proposed functional, two main relations are identified. First, the chemical potential appears as the opposite of the Lagrange multiplier that allows to enforce the balance of mass and natural boundary conditions. Second, a conjugate relation is found for a given substance between its mass flux and the opposite of the gradient of its chemical potential. Furthermore, to reduce the number of variables of the initial variational principle, a field reduction is applied, reaching the model presented by Miehe et al. (Int J Numer Methods Eng 99(10):737, 2014. https://doi.org/10.1002/nme.4700) for Fickean diffusion. Nevertheless, the aforementioned relations cannot be derived from the reduced model. Finally, a numerical implementation is presented for completeness where we compare the performance of the proposed formulations against the usual weak form.

对于扩散的初始边值问题，我们提出了两个多场和一个单场变分原理。化学势和浓度在多场配方中显示为共轭变量。提议的配方的主要重要性是用于生成变体原理的方法，其中通用标准材料的框架本构关系用于本构律，而自然边界条件和质量平衡则用作优化问题的约束。这种方法允许以一般方式推导多物理场问题的这种原理。给出了详细的推导和分析公式，使用菲克定律和对数质量作用定律可以看出它们与问题的最常见的强形式和弱形式等效。从相对于最初提出的函数的质量通量的平稳性条件，可以确定两个主要关系。首先，化学势似乎与拉格朗日乘数相反，后者可以加强质量和自然边界条件之间的平衡。第二，对于给定物质，在其质量通量和其化学势梯度的反面之间找到​​共轭关系。此外，为了减少初始变分原理的变量数量，应用了场减小法，达到了Miehe等人提出的模型。（Int J Numer Methods Eng 99（10）：737，2014.https：//doi.org/10.1002/nme.4700）用于Fickean扩散。但是，上述关系不能从简化模型中得出。最后，为了完整起见，提出了一个数值实现方案，其中我们将所提出的配方的性能与通常的弱形式进行了比较。（Int J Numer Methods Eng 99（10）：737，2014.https：//doi.org/10.1002/nme.4700）用于Fickean扩散。但是，上述关系不能从简化模型中得出。最后，为了完整起见，提出了一个数值实现方案，其中我们将所提出的配方的性能与通常的弱形式进行了比较。（Int J Numer Methods Eng 99（10）：737，2014.https：//doi.org/10.1002/nme.4700）用于Fickean扩散。但是，上述关系不能从简化模型中得出。最后，为了完整起见，提出了一个数值实现方案，其中我们将所提出的配方的性能与通常的弱形式进行了比较。